Include the names of your collaborators here.
This homework assignment is focused on model complexity and the influence of the prior regularization strength. You will fit non-Bayesian and Bayesian linear models, compare them, and make predictions to visualize the trends. You will use multiple prior strengths to study the impact on the coefficient posteriors and on the posterior predictive distributions.
You are also introduced to non-Bayesian regularization with Lasso
regression via the glmnet package. If you do not have
glmnet installed please download it before starting the
assignment.
IMPORTANT: The RMarkdown assumes you have downloaded the data set (CSV file) to the same directory you saved the template Rmarkdown file. If you do not have the CSV files in the correct location, the data will not be loaded correctly.
Certain code chunks are created for you. Each code chunk has
eval=FALSE set in the chunk options. You
MUST change it to be eval=TRUE in order
for the code chunks to be evaluated when rendering the document.
You are free to add more code chunks if you would like.
This assignment will use packages from the tidyverse
suite as well as the coefplot package. Those packages are
imported for you below.
library(tidyverse)
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.4.0 ✔ purrr 1.0.1
## ✔ tibble 3.1.8 ✔ dplyr 1.0.10
## ✔ tidyr 1.2.1 ✔ stringr 1.5.0
## ✔ readr 2.1.3 ✔ forcats 0.5.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
library(coefplot)
This assignment also uses the splines and
MASS packages. Both are installed with base R
and so you do not need to download any additional packages to complete
the assignment.
The last question in the assignment uses the glmnet
package. As stated previously, please download and install
glmnet if you do not currently have it.
You will fit and compare 6 models of varying
complexity using non-Bayesian methods. The unknown
parameters will be be estimated by finding their Maximum Likelihood
Estimates (MLE). You are allowed to use the lm() function
for this problem.
The data are loaded in the code chunk and a glimpse is shown for you
below. There are 2 continuous inputs, x1 and
x2, and a continuous response y.
hw_file_path <- 'hw08_data.csv'
df <- readr::read_csv(hw_file_path, col_names = TRUE)
## Rows: 100 Columns: 3
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (3): x1, x2, y
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
df %>% glimpse()
## Rows: 100
## Columns: 3
## $ x1 <dbl> -0.30923281, 0.63127211, -0.68276690, 0.26930562, 0.37252021, 1.296…
## $ x2 <dbl> 0.308779853, -0.547919793, 2.166449412, 1.209703658, 0.785485991, -…
## $ y <dbl> 0.43636596, 1.37562976, -0.84366730, -0.43080811, 0.77456951, 1.361…
Create a scatter plot between the response, y,
and each input using ggplot().
Based on the visualizations, do you think there are trends between either input and the response?
p1 <- ggplot(df, aes(x = x1, y = y)) +
geom_point(size = 2, color = "red") +
labs(title = "Scatter plot of y vs x1", x = "x1", y = "y")
p2 <- ggplot(df, aes(x = x2, y = y)) +
geom_point(size = 2, color = "blue") +
labs(title = "Scatter plot of y vs x2", x = "x2", y = "y")
library(patchwork)
p1 + p2
There is basically no connection between x1 and y, and the points are very scattered. In the x2 diagram, we can clearly see that the closer x2 is to 0, the greater the y value; the closer to both ends, the smaller the y value.
You will fit multiple models of varying complexity in this problem. You will start with linear additive features which add the effect of one input with the other. Your model therefore controls for both inputs.
Fit a model with linear additive features to predict the
response, y. Use the formula interface and the
lm() function to fit the model. Assign the result to the
mod01 object.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
mod01 <- lm( y ~ x1 + x2, df )
coefplot(mod01)
It can be seen from the figure that the confidence interval of x2 covers both negative and positive values, so it is not credible, and the value of x1 covers all positive values, so it has statistically significant.
As discussed in lecture, we can derive features from inputs. We have worked with polynomial features and spline-based features in previous assignments. Features can also be derived as the products between different inputs. A feature calculated as the product of multiple inputs is usually referred to as the interaction between those inputs.
In the formula interface, a product of two inputs is denoted by the
:. And so if we want to include just the multiplication of
x1 and x2 in a model we would type,
x1:x2. We can then include main-effect
terms by including the additive features within the formula. Thus, the
formula for a model with additive features and the interaction between
x1 and x2 is:
y ~ x1 + x2 + x1:x2
However, the formula interface provides a short-cut to create main
effects and interaction features. In the formula interface, the
* operator will generate all main-effects and all
interactions for us.
Fit a model with all main-effect and all-interaction features
between x1 and x2 using the short-cut
* operator within the formula interface. Assign the result
to the mod02 object.
Visualize the coefficient summaries with the
coefplot() function. How many features are present in the
model? Are any of the features statistically significant?
mod02 <- lm( y ~ x1*x2, df )
coefplot(mod02)
3 features are present in the model. It can be seen from the figure that the confidence intervals of x1, x2, and x1:x2 have not crossed the 0 value, so they all have statistically significant.
The * operator will interact more than just inputs. We
can interact expressions or groups of features together. To interact one
group of features by another group of features, we just need to enclose
each group within parenthesis, (), and separate them by the
* operator. The line of code below shows how this works
with the <expression 1> and
<expression 2> as place holders for any expression we
want to use.
(<expression 1>) * (<expression 2>)
Fit a model which interacts linear and quadratic features
from x1 with linear and quadratic features from
x2. Assign the result to the mod03
object.
Visualize the coefficient summaries with the
coefplot() function. How many features are present in the
model? Are any of the features statistically significant?
HINT: Remember to use the I() function when
typing polynomials in the formula interface.
mod03 <- lm( y ~ (x1 + I(x1^2)) * (x2 + I(x2^2)), df )
coefplot(mod03)
As can be seen from the figure, there are eight features in total. Same as above, only the quadratic feature associated with x2 has statistically significant.
Let’s now try a more complicated model.
Fit a model which interacts linear, quadratic, cubic, and
quartic (4th degree) polynomial features from x1 with
linear, quadratic, cubic, and quartic (4th degree) polynomial features
from x2. Assign the result to the mod04
object.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
mod04 <- lm( y ~ (x1 + I(x1^2) + I(x1^3) + I(x1^4)) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), df )
coefplot(mod04)
Same as above, none of the features has statistically significant.
Let’s try using spline based features. We will use a high
degree-of-freedom natural spline applied to x1 and interact
those features with polynomial features derived from
x2.
Fit a model which interacts a 12 degree-of-freedom natural
(DOF) spline from x1 with linear and quadrtic polyonomial
features from x2. Assign the result to
mod05.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
mod05 <- lm(y ~ splines::ns(x1, 12) * (x2 + I(x2^2)),df)
coefplot(mod05)
Same as above, none of the features has statistically significant.
Let’s fit one final model.
Fit a model which interacts a 12 degree-of-freedom natural
spline from x1 with linear, quadrtic, cubic, and quartic
(4th degree) polyonomial features from x2. Assign the
result to mod05.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
mod06 <- lm(y ~ splines::ns(x1, 12) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), df)
coefplot(mod06)
Same as above, none of the features has statistically significant.
Now that you have fit multiple models of varying complexity, it is time to identify the best performing model.
Identify the best model considering training set only performance metrics. Which model is best according to R-squared? Which model is best according to AIC? Which model is best according to BIC?
HINT: The brooom::glance() function can be
helpful here. The broom package is installed with
tidyverse and so you should have it already.
# fit models and extract performance metrics
models <- list(mod01, mod02, mod03, mod04, mod05, mod06)
metrics <- lapply(models, function(mod) {
broom::glance(mod)
})
# extract performance metrics for each model
r_squared <- sapply(metrics, "[[", "r.squared")
AIC <- sapply(metrics, "[[", "AIC")
BIC <- sapply(metrics, "[[", "BIC")
# print metrics for each model
for (i in seq_along(models)) {
cat(sprintf("Model %02d:\n", i))
cat(sprintf("R-squared: %f\n", r_squared[i]))
cat(sprintf("AIC: %f\n", AIC[i]))
cat(sprintf("BIC: %f\n", BIC[i]))
cat("\n")
}
## Model 01:
## R-squared: 0.059447
## AIC: 284.653988
## BIC: 295.074669
##
## Model 02:
## R-squared: 0.113010
## AIC: 280.790492
## BIC: 293.816342
##
## Model 03:
## R-squared: 0.546542
## AIC: 223.697344
## BIC: 249.749045
##
## Model 04:
## R-squared: 0.598728
## AIC: 243.470980
## BIC: 311.205405
##
## Model 05:
## R-squared: 0.699279
## AIC: 242.625329
## BIC: 346.832136
##
## Model 06:
## R-squared: 0.781894
## AIC: 262.505077
## BIC: 434.446310
# identify best performing model according to R-squared
best_r_squared <- which.max(r_squared)
cat(sprintf("The best model according to R-squared is Model %02d\n", best_r_squared))
## The best model according to R-squared is Model 06
# identify best performing model according to AIC
best_AIC <- which.min(AIC)
cat(sprintf("The best model according to AIC is Model %02d\n", best_AIC))
## The best model according to AIC is Model 03
# identify best performing model according to BIC
best_BIC <- which.min(BIC)
cat(sprintf("The best model according to BIC is Model %02d\n", best_BIC))
## The best model according to BIC is Model 03
The best model according to R-squared has the highest value, mod06. The best model according to AIC has the lowest value, mod03. The best model according to BIC has the lowest value, mod03.
You will define a prediction or visualization test grid. This grid
will allow you to visualize behavior with respect to x1 for
multiple values of x2.
Create a grid of input values where x1 consists
of 101 evenly spaced points between -3.2 and 3.2 and x2 is
9 evenly spaced points between -3 and 3. The expand.grid()
function is started for you and the data type conversion is provided to
force the result to be a tibble.
viz_grid <- expand.grid(x1 = seq(-3.2, 3.2, length.out = 101),
x2 = seq(-3, 3, length.out = 9),
KEEP.OUT.ATTRS = FALSE,
stringsAsFactors = FALSE) %>%
as.data.frame() %>% tibble::as_tibble()
You will make predictions for each of the models and visualize their
trends. A function, tidy_predict(), is created for you
which assembles the predicted mean trend, the confidence interval, and
the prediction interval into a tibble for you. The result
include the input values to streamline making the visualizations.
tidy_predict <- function(mod, xnew)
{
pred_df <- predict(mod, xnew, interval = "confidence") %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred = fit, ci_lwr = lwr, ci_upr = upr) %>%
bind_cols(predict(mod, xnew, interval = 'prediction') %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred_lwr = lwr, pred_upr = upr))
xnew %>% bind_cols(pred_df)
}
The first argument to the tidy_predict() function is a
lm() model object and the second argument is new or test
dataframe of inputs. When working with lm() and its
predict() method, the functions will create the test design
matrix consistent with the training design basis. It does so via the
model object’s formula which is contained within the lm()
model object. The lm() object therefore takes care of the
heavy lifting for us!
Make predictions with each of the six models you fit in
Problem 01 using the visualization grid, viz_grid. The
predictions should be assigned to the variables pred_lm_01
through pred_lm_06 where the number is consistent with the
model number fit previously.
pred_lm_01 <- tidy_predict(mod01, viz_grid)
pred_lm_02 <- tidy_predict(mod02, viz_grid)
pred_lm_03 <- tidy_predict(mod03, viz_grid)
pred_lm_04 <- tidy_predict(mod04, viz_grid)
pred_lm_05 <- tidy_predict(mod05, viz_grid)
pred_lm_06 <- tidy_predict(mod06, viz_grid)
You will now visualize the predictive trends and the confidence and
prediction intervals for each model. The pred column in of
each pred_lm_ objects is the predictive mean trend. The
ci_lwr and ci_upr columns are the lower and
upper bounds of the confidence interval, respectively. The
pred_lwr and pred_upr columns are the lower
and upper bounds of the prediction interval, respectively.
You will use ggplot() to visualize the predictions. You
will use geom_line() to visualize the mean trend and
geom_ribbon() to visualize the uncertainty intervals.
Visualize the predictions of each model on the visualization
grid. Pipe the pred_lm_ object to ggplot() and
map the x1 variable to the x-aesthetic. Add three geometric
object layers. The first and second layers are each
geom_ribbon() and the third layer is
geom_line(). In the geom_line() layer map the
pred variable to the y aesthetic. In the first
geom_ribbon() layer, map pred_lwr and
pred_upr to the ymin and ymax
aesthetics, respectively. Hard code the fill to be orange
in the first geom_ribbon() layer (outside the
aes() call). In the second geom_ribbon()
layer, map ci_lwr and ci_upr to the
ymin and ymax aesthetics, respectively. Hard
code the fill to be grey in the second
geom_ribbon() layer (outside the aes() call).
Include facet_wrap() with the facets with controlled by the
x2 variable.
To help compare the visualizations across models include a
coord_cartesian() layer with the ylim argument
set to c(-7,7).
Each model’s prediction visualization should be created in a separate code chunk.
ggplot(pred_lm_01 ,mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~x2)+
coord_cartesian(ylim = c(-7, 7))
ggplot(pred_lm_02 ,mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~x2)+
coord_cartesian(ylim = c(-7, 7))
ggplot(pred_lm_03 ,mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~x2)+
coord_cartesian(ylim = c(-7, 7))
ggplot(pred_lm_04 ,mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~x2)+
coord_cartesian(ylim = c(-7, 7))
ggplot(pred_lm_05 ,mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~x2)+
coord_cartesian(ylim = c(-7, 7))
ggplot(pred_lm_06 ,mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr),
fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~x2)+
coord_cartesian(ylim = c(-7, 7))
### 2d)
Do you feel the predictions are consistent with the model performance rankings based on AIC/BIC? What is the defining characteristic of the models considered to be the worst by AIC/BIC?
What do you think?
I think the predictions doesn’t consistent with the model performance
rankings based on AIC/BIC. We know the black line represents the trend,
the gray represents the confidence interval, and the orange represents
the prediction interval. From model 1 to model 6, we can see that the
trend is getting less and less obvious, the confidence interval is
getting bigger and bigger, but the prediction interval is shrinking.
AIC and BIC are statistical measures used to compare the goodness of fit of different models. The worst models according to AIC/BIC are those with high AIC/BIC values, indicating that they have poor fit to the data and are too complex relative to the amount of data available.
In simpler terms, AIC/BIC penalizes models that are overly complicated or that do not fit the data well. The more complex a model is, the more likely it is to overfit the data, which means it fits the noise in the data rather than the underlying pattern.
Now that you have fit non-Bayesian linear models with maximum likelihood estimation, it is time to use Bayesian models to understand the influence of the prior on the model behavior.
Regardless of your answers in Problem 02 you will only work with model 3 and model 6 in this problem.
You will perform the Bayesian analysis using the Laplace Approximation just as you did in the previous assignment. You will define the log-posterior function just as you did in the previous assignment and so before doing so you must create the list of required information. This list will include the observed response, the design matrix, and the prior specification. You will use independent Gaussian priors on the regression parameters with a shared prior mean and shared prior standard deviation. You will use an Exponential prior on the unknown likelihood noise (the \(\sigma\) parameter).
Complete the two code chunks below. In the first, create the
design matrix following mod03’s formula, and assign the
object to the X03 variable. Complete the
info_03_weak list by assigning the response to
yobs and the design matrix to design_matrix.
Specify the shared prior mean, mu_beta, to be 0, the shared
prior standard deviation, tau_beta, as 50, and the rate
parameter on the noise, sigma_rate, to be 1.
Complete the second code chunk with the same prior
specification. The second code chunk however requires that you create
the design matrix associated with mod06’s formula and
assign the object to the X06 variable. Assign
X06 to the design_matrix field of the
info_06_weak list.
X03 <- model.matrix(y ~ (x1 + I(x1^2)) * (x2 + I(x2^2)), df)
info_03_weak <- list(
yobs = df$y,
design_matrix = X03,
mu_beta = 0,
tau_beta = 50,
sigma_rate = 1
)
X06 <- model.matrix(y ~ splines::ns(x1, 12) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), df)
info_06_weak <- list(
yobs = df$y,
design_matrix = X06,
mu_beta = 0,
tau_beta = 50,
sigma_rate = 1
)
You will now define the log-posterior function
lm_logpost(). You will continue to use the
log-transformation on \(\sigma\), and
so you will actually define the log-posterior in terms of the mean trend
\(\boldsymbol{\beta}\)-parameters and
the unbounded noise parameter, \(\varphi =
\log\left[\sigma\right]\).
The comments in the code chunk below tell you what you need to fill
in. The unknown parameters to learn are contained within the first input
argument, unknowns. You will assume that the unknown \(\boldsymbol{\beta}\)-parameters are listed
before the unknown \(\varphi\)
parameter in the unknowns vector. You must specify the
number of \(\boldsymbol{\beta}\)
parameters programmatically to allow scaling up your function to an
arbitrary number of unknowns. You will assume that all variables
contained in the my_info list (the second argument to
lm_logpost()) are the same fields in the
info_03_weak list you defined in Problem 3a).
Define the log-posterior function by completing the code
chunk below. You must calculate the mean trend, mu, using
matrix math between the design matrix and the unknown \(\boldsymbol{\beta}\) column
vector.
HINT: This function should look very famaliar…
lm_logpost <- function(unknowns, my_info)
{
# specify the number of unknown beta parameters
length_beta <- ncol(my_info$design_matrix)
# extract the beta parameters from the `unknowns` vector
beta_v <- unknowns[1:length_beta]
# extract the unbounded noise parameter, varphi
lik_varphi <- unknowns[length_beta + 1]
# back-transform from varphi to sigma
lik_sigma <- exp(lik_varphi)
# extract design matrix
X <- my_info$design_matrix
# calculate the linear predictor
mu <- as.vector( X %*% as.matrix(beta_v) )
# evaluate the log-likelihood
log_lik <- sum(dnorm(x = my_info$yobs,
mean = mu,
sd = lik_sigma,
log = TRUE))
# evaluate the log-prior
log_prior_beta <- sum(dnorm(x = beta_v,
mean = my_info$mu_beta,
sd = my_info$tau_beta,
log = TRUE))
log_prior_sigma <- dexp(x = lik_sigma,
rate = my_info$sigma_rate,
log = TRUE)
# add the mean trend prior and noise prior together
log_prior <- log_prior_beta + log_prior_sigma
# account for the transformation
log_derive_adjust <- lik_varphi
# sum together
value1<-log_lik + log_prior + log_derive_adjust
value1
}
The my_laplace() function is defined for you in the code
chunk below. This function executes the laplace approximation and
returns the object consisting of the posterior mode, posterior
covariance matrix, and the log-evidence.
my_laplace <- function(start_guess, logpost_func, ...)
{
# code adapted from the `LearnBayes`` function `laplace()`
fit <- optim(start_guess,
logpost_func,
gr = NULL,
...,
method = "BFGS",
hessian = TRUE,
control = list(fnscale = -1, maxit = 1001))
mode <- fit$par
post_var_matrix <- -solve(fit$hessian)
p <- length(mode)
int <- p/2 * log(2 * pi) + 0.5 * log(det(post_var_matrix)) + logpost_func(mode, ...)
# package all of the results into a list
list(mode = mode,
var_matrix = post_var_matrix,
log_evidence = int,
converge = ifelse(fit$convergence == 0,
"YES",
"NO"),
iter_counts = as.numeric(fit$counts[1]))
}
Execute the Laplace Approximation for the model 3 formulation
and the model 6 formulation. Assign the model 3 result to the
laplace_03_weak object, and assign the model 6 result to
the laplace_06_weak object. Check that the optimization
scheme converged.
laplace_03_weak <- my_laplace(rep(0, ncol(X03)+1), lm_logpost, info_03_weak)
laplace_03_weak
## $mode
## [1] 0.665297498 0.164476875 -0.160380913 -0.051976413 -0.556454210
## [6] 0.122777593 -0.082522064 0.005332016 0.020385343 -0.398793318
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.382473e-02 -2.857502e-04 -6.974580e-03 3.995613e-04 -5.817298e-03
## [2,] -2.857502e-04 7.785753e-03 1.697328e-04 8.041568e-04 -5.563936e-04
## [3,] -6.974580e-03 1.697328e-04 7.581322e-03 1.620125e-04 3.406310e-03
## [4,] 3.995613e-04 8.041568e-04 1.620125e-04 8.160422e-03 5.829597e-04
## [5,] -5.817298e-03 -5.563936e-04 3.406310e-03 5.829597e-04 5.670858e-03
## [6,] 2.435735e-03 -2.044408e-05 1.411797e-04 6.038786e-04 -1.933614e-03
## [7,] -1.669196e-03 -2.966249e-03 6.048931e-04 -1.077128e-03 1.525096e-03
## [8,] 1.328275e-03 1.841367e-04 -5.911773e-04 -2.460710e-03 -1.598468e-03
## [9,] 2.527267e-03 3.147474e-04 -3.492379e-03 -9.078644e-04 -2.395812e-03
## [10,] -8.813906e-09 4.091253e-08 -5.264766e-08 3.626197e-08 8.444137e-08
## [,6] [,7] [,8] [,9] [,10]
## [1,] 2.435735e-03 -1.669196e-03 1.328275e-03 2.527267e-03 -8.813906e-09
## [2,] -2.044408e-05 -2.966249e-03 1.841367e-04 3.147474e-04 4.091253e-08
## [3,] 1.411797e-04 6.048931e-04 -5.911773e-04 -3.492379e-03 -5.264766e-08
## [4,] 6.038786e-04 -1.077128e-03 -2.460710e-03 -9.078644e-04 3.626197e-08
## [5,] -1.933614e-03 1.525096e-03 -1.598468e-03 -2.395812e-03 8.444137e-08
## [6,] 1.216999e-02 -4.522317e-03 4.252590e-03 -2.772348e-03 -1.313317e-07
## [7,] -4.522317e-03 4.979084e-03 -2.548218e-03 8.311808e-04 -1.409092e-08
## [8,] 4.252590e-03 -2.548218e-03 4.465146e-03 -4.413637e-04 -8.913149e-08
## [9,] -2.772348e-03 8.311808e-04 -4.413637e-04 3.461386e-03 -1.730025e-08
## [10,] -1.313317e-07 -1.409092e-08 -8.913149e-08 -1.730025e-08 4.999773e-03
##
## $log_evidence
## [1] -164.7933
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 67
laplace_06_weak <- my_laplace(rep(0, ncol(X06)+1), lm_logpost, info_06_weak)
laplace_06_weak
## $mode
## [1] -6.5207834 3.3423300 8.9867547 4.4301616 8.8854915 6.7351266
## [7] 7.5251127 7.1970111 7.3760587 4.7287927 4.7769825 16.5387116
## [13] 2.3736018 0.5865565 7.8684602 -1.0303226 -3.1185282 0.3019329
## [19] -3.1150237 2.4434748 -1.9024975 1.1105308 -1.0262699 -1.1203347
## [25] -2.6808387 0.6611046 -5.3044978 5.1906671 13.0976539 1.0847170
## [31] -17.1669832 3.1155370 -13.0459447 -5.4077670 -10.4130431 -10.9819910
## [37] -8.0175592 10.2152376 -11.9892792 -23.1224440 0.4810989 -1.5515068
## [43] 2.6250521 -2.5519567 2.4268546 -0.6329237 1.9752677 2.2250425
## [49] -4.7658272 17.1636200 -5.0155458 -23.6570738 -38.3167356 0.5988171
## [55] 5.3163663 0.4063300 3.9179374 2.7775225 2.5726489 7.3743241
## [61] -7.0403689 -6.7706288 4.4624827 22.3899530 22.7447404 -0.7577507
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 13.952023525 -11.100071970 -15.42731051 -12.793671476 -14.64476686
## [2,] -11.100071970 10.191999074 11.51181182 10.730980494 11.34547555
## [3,] -15.427310511 11.511811819 17.66158415 13.674469732 16.44980855
## [4,] -12.793671476 10.730980494 13.67446973 12.546784353 12.98690314
## [5,] -14.644766864 11.345475554 16.44980855 12.986903143 15.82076980
## [6,] -13.585726001 10.991299618 14.86246020 12.788665856 13.91758005
## [7,] -14.168865584 11.146368668 15.77859636 12.761046161 15.09584219
## [8,] -13.862804719 11.090298473 15.27565337 12.820168943 14.44177219
## [9,] -14.030905448 11.131504607 15.54228907 12.809663349 14.78206988
## [10,] -13.774947743 10.995569290 15.20193130 12.671585604 14.42478163
## [11,] -7.313207088 6.218197212 7.86540220 6.852250971 7.60652331
## [12,] -30.174218176 22.998573806 33.93075302 27.274033902 31.87973501
## [13,] -3.974803841 3.789010650 4.04251707 3.901970686 4.01007045
## [14,] -17.615596135 17.055646467 17.69342268 16.898695451 18.07847003
## [15,] -5.446445631 1.272459352 7.68034937 4.201892441 6.15206640
## [16,] 9.909347166 -9.846709771 -9.77314875 -9.635065995 -10.10646735
## [17,] -0.760357646 2.008580493 0.11417643 1.014224941 0.61865817
## [18,] 17.062201666 -16.640288767 -17.06249770 -16.478484534 -17.47520120
## [19,] 18.063053518 -17.163620021 -18.43800514 -16.894783675 -18.67225376
## [20,] 16.900376852 -16.935373508 -16.45467248 -17.287426906 -16.94148139
## [21,] 17.803314770 -17.044642969 -18.06391259 -16.689626285 -18.41958706
## [22,] 17.514141201 -17.121253046 -17.42503081 -17.233894306 -17.70430506
## [23,] 18.023613431 -17.131987524 -18.36993574 -16.832465310 -18.81911933
## [24,] 17.611233328 -17.045045827 -17.69105478 -16.891836878 -18.11768202
## [25,] 17.580834822 -17.064126218 -17.62546414 -16.925701966 -17.96155655
## [26,] 17.661788147 -17.032993359 -17.78361907 -16.909562551 -18.17838954
## [27,] 10.553316210 -9.977748804 -10.74542225 -10.027585780 -10.86508522
## [28,] 34.931539638 -34.386321132 -34.75744987 -33.700374435 -35.74475091
## [29,] 5.636228740 -5.495633914 -5.57122380 -5.581851416 -5.73948627
## [30,] 0.060843800 -0.848096827 0.36859704 -0.992421705 0.44122207
## [31,] 8.034716837 -1.016028350 -12.47647017 -4.590978893 -9.80347932
## [32,] 2.164668707 -0.646282515 -2.29441716 -4.351439741 -1.27075065
## [33,] 7.204258377 -1.768392850 -10.40797502 -4.426896734 -9.11763426
## [34,] 4.931595281 -1.279583946 -6.70572910 -4.757804052 -4.73224151
## [35,] 5.563769227 -1.163388628 -8.02276622 -3.677553480 -6.80523620
## [36,] 5.359732430 -1.393052103 -7.40930013 -4.527727692 -5.70727983
## [37,] 5.543082611 -1.242798191 -7.89532757 -4.019390264 -6.48015276
## [38,] 4.363227292 -0.582645448 -6.32063716 -3.447528750 -4.88921614
## [39,] 1.869597900 -0.971387570 -2.37490859 -1.710109797 -1.96746077
## [40,] 14.397628409 -2.089338146 -20.98970934 -10.291842683 -16.67132173
## [41,] -2.871925691 1.499579480 3.63382737 1.993711673 3.41278910
## [42,] -7.615774168 8.627972507 6.99019434 7.821875043 7.53596950
## [43,] -10.814243986 10.349967628 10.87853353 10.229020981 11.15418057
## [44,] -8.752978088 9.508262023 7.98912305 9.482269885 8.49766975
## [45,] -10.431792541 10.003006379 10.57371519 9.738729826 10.86219183
## [46,] -9.703240201 9.851933770 9.38338898 9.840229917 9.59381004
## [47,] -10.344041265 9.943968854 10.47093537 9.630485203 10.87829920
## [48,] -10.018976568 9.856687034 9.96560042 9.582264479 10.37292986
## [49,] -9.387242931 9.591909626 9.03982573 9.477497103 9.28934986
## [50,] -10.734554703 10.301031376 10.85881561 10.170483670 11.12118330
## [51,] -4.468150931 4.384610681 4.41690719 4.379748959 4.51626077
## [52,] -19.658192992 20.749781925 18.58448381 19.752455033 19.72043001
## [53,] 4.372884416 -2.984250336 -5.31552872 -3.298744407 -4.91620471
## [54,] 1.535423525 -1.452910157 -1.59815593 -1.226075491 -1.70195805
## [55,] 0.362426805 -2.304631885 0.83012982 -1.199438822 0.04752874
## [56,] 1.418162586 -2.046590874 -1.28680037 -0.691658347 -1.68421831
## [57,] 0.471947481 -1.964636900 0.37502225 -1.121071618 -0.04159788
## [58,] 0.497316621 -1.812370816 0.15608641 -0.503073438 -0.57823077
## [59,] 1.161796567 -2.238507555 -0.60895831 -1.487896734 -0.94300406
## [60,] 0.546053548 -1.774121487 0.06451224 -0.653356343 -0.55825062
## [61,] 1.401289228 -2.520419377 -0.82381599 -1.666557529 -1.18153546
## [62,] 1.212602611 -2.254587118 -0.71282227 -1.299923097 -1.17491896
## [63,] 0.319914612 -0.419470620 -0.27223852 -0.256600385 -0.34106495
## [64,] 0.298098338 -4.847982014 2.15128985 -1.725308394 0.47290254
## [65,] -2.814844801 2.528698939 3.04188180 2.497026481 2.95916485
## [66,] 0.008362952 -0.003691296 -0.01102439 -0.006423542 -0.00944052
## [,6] [,7] [,8] [,9] [,10]
## [1,] -13.585726001 -14.168865584 -13.86280472 -14.030905448 -13.77494774
## [2,] 10.991299618 11.146368668 11.09029847 11.131504607 10.99556929
## [3,] 14.862460201 15.778596363 15.27565337 15.542289066 15.20193130
## [4,] 12.788665856 12.761046161 12.82016894 12.809663349 12.67158560
## [5,] 13.917580047 15.095842188 14.44177219 14.782069879 14.42478163
## [6,] 13.775049903 13.402205037 13.72234630 13.549238277 13.47446204
## [7,] 13.402205037 15.291453973 13.39887785 14.604481414 13.80280589
## [8,] 13.722346301 13.398877851 14.64531691 13.469969537 13.92128767
## [9,] 13.549238277 14.604481414 13.46996954 14.738821918 13.39170517
## [10,] 13.474462042 13.802805890 13.92128767 13.391705169 14.89338252
## [11,] 7.136074173 7.515326692 7.11561964 7.689680419 6.36975252
## [12,] 29.274678961 30.645206790 30.06331337 30.123032167 30.25350429
## [13,] 4.024850709 3.733480535 4.31697573 3.324646765 5.37213896
## [14,] 17.504492114 17.641682105 17.67198946 17.672228449 17.37718612
## [15,] 4.951776497 5.777268105 5.25292932 5.526915831 5.35286258
## [16,] -9.887540123 -9.890787396 -9.95780129 -9.933871841 -9.77922883
## [17,] 0.893033404 0.668828355 0.82538395 0.744168111 0.76096423
## [18,] -16.980973209 -17.062719287 -17.12308339 -17.108300763 -16.84652180
## [19,] -17.832013290 -18.210241432 -18.07433576 -18.153082540 -17.78700563
## [20,] -17.112938077 -16.638582788 -17.07199770 -16.881820801 -16.73827767
## [21,] -17.550214353 -17.974048985 -17.80769332 -17.896712072 -17.53293533
## [22,] -17.695573658 -17.273005459 -17.60605072 -17.571330333 -17.19054894
## [23,] -17.605847314 -18.319284295 -18.05417993 -18.078440938 -17.90147620
## [24,] -17.286634373 -18.619916370 -16.33725125 -18.462708209 -16.92848245
## [25,] -17.748465404 -16.440113380 -19.16443924 -16.643883345 -18.18149725
## [26,] -17.357855091 -18.539120821 -16.62418254 -18.491761814 -15.20933589
## [27,] -10.561538951 -10.044115868 -11.32038341 -10.258619958 -10.97341670
## [28,] -34.743772554 -35.168399471 -34.72374011 -35.036542486 -35.21641399
## [29,] -5.492536272 -6.401230617 -4.53689579 -5.882530818 -6.46356735
## [30,] -0.248511074 0.086034082 -0.08678867 -0.011087859 -0.17025302
## [31,] -6.870155132 -8.902277401 -7.60753376 -8.256082183 -7.76274320
## [32,] -2.942039385 -1.498826006 -2.40867410 -1.997451792 -2.27790963
## [33,] -5.754314481 -8.242518184 -6.69371365 -7.457099298 -6.97693326
## [34,] -5.721730111 -4.291922772 -5.18528365 -4.791602793 -4.91201574
## [35,] -4.196455699 -7.350850015 -4.33838798 -6.189075443 -5.24951931
## [36,] -5.519432032 -3.931332040 -7.45481633 -4.205634110 -5.33771465
## [37,] -4.580615005 -7.303722037 -3.42085953 -7.988469440 -4.88327389
## [38,] -4.001863519 -4.589196046 -4.18314859 -3.463293773 -9.06232012
## [39,] -1.759632404 -1.875977004 -1.89143480 -2.497555199 0.74904233
## [40,] -12.972432066 -15.165030433 -14.19993422 -13.599142561 -16.30587247
## [41,] 2.483505642 3.442778444 2.24266129 5.154444444 -2.86598201
## [42,] 7.730113898 7.509047744 7.69702376 7.611195427 7.54945222
## [43,] 10.703403141 10.865701326 10.83717691 10.857822762 10.65533607
## [44,] 8.993601348 8.514641536 8.88481913 8.722099157 8.70254305
## [45,] 10.272885212 10.529351827 10.44416553 10.478794166 10.27192386
## [46,] 9.974975619 9.462673695 9.74556772 9.771030368 9.43859381
## [47,] 10.046678872 10.451442528 10.50659963 10.257279106 10.40869949
## [48,] 9.585866449 11.541778526 8.05316218 11.303350307 9.19991951
## [49,] 10.042289218 6.917202449 12.53849870 6.966960124 11.45790957
## [50,] 10.383325486 11.884212300 9.44786014 11.900578520 5.27823975
## [51,] 4.640311526 3.583303271 5.56685496 4.001034683 6.72955764
## [52,] 19.777828417 20.012648416 18.96861661 18.437786848 25.63081767
## [53,] -4.285931613 -3.034535033 -6.52951777 -6.202327205 4.66937778
## [54,] -1.484463653 -1.574396291 -1.53790802 -1.552794035 -1.48936071
## [55,] -0.683813279 -0.118029472 -0.49442410 -0.308366484 -0.40212028
## [56,] -1.150518692 -1.644276770 -1.34476572 -1.474845876 -1.38510429
## [57,] -0.882016061 -0.170969082 -0.62676673 -0.411505341 -0.48663010
## [58,] -0.212028382 -0.719696924 -0.45588965 -0.509407869 -0.57661444
## [59,] -1.552513220 -0.635402585 -1.54453455 -1.004302087 -1.08226578
## [60,] -0.307801307 -1.650641954 0.96869667 -1.303454470 -1.01300300
## [61,] -1.900867552 0.200249724 -3.52525246 0.003007229 -0.03916132
## [62,] -1.186052187 -1.637684163 -0.69962166 -1.825943322 1.78667677
## [63,] -0.402568011 0.065721902 -0.81266430 -0.030800667 -1.61302786
## [64,] -0.815454287 -0.250379214 0.01083985 0.185097243 -3.18508068
## [65,] 2.866307410 2.003558401 4.06334571 3.306508599 -1.11195255
## [66,] -0.007718723 -0.008465339 -0.00874500 -0.009037369 -0.00486937
## [,11] [,12] [,13] [,14] [,15]
## [1,] -7.313207088 -30.174218176 -3.974803841 -17.61559614 -5.4464456
## [2,] 6.218197212 22.998573806 3.789010650 17.05564647 1.2724594
## [3,] 7.865402197 33.930753024 4.042517069 17.69342268 7.6803494
## [4,] 6.852250971 27.274033902 3.901970686 16.89869545 4.2018924
## [5,] 7.606523309 31.879735005 4.010070450 18.07847003 6.1520664
## [6,] 7.136074173 29.274678961 4.024850709 17.50449211 4.9517765
## [7,] 7.515326692 30.645206790 3.733480535 17.64168210 5.7772681
## [8,] 7.115619638 30.063313368 4.316975726 17.67198946 5.2529293
## [9,] 7.689680419 30.123032167 3.324646765 17.67222845 5.5269158
## [10,] 6.369752525 30.253504287 5.372138964 17.37718612 5.3528626
## [11,] 5.121309926 14.394038662 -0.100695563 10.55944150 1.7010489
## [12,] 14.394038662 67.935654304 11.312037674 34.81134234 14.6764897
## [13,] -0.100695563 11.312037674 9.029255245 7.11261362 -0.4484109
## [14,] 10.559441499 34.811342342 7.112613618 64.85442118 -27.7611279
## [15,] 1.701048871 14.676489674 -0.448410879 -27.76112786 37.9286779
## [16,] -6.105078647 -19.177513427 -4.218990758 -38.37243595 16.2347846
## [17,] 0.873526226 0.444241164 1.092684429 17.38655308 -17.8226954
## [18,] -10.237869928 -33.685761035 -6.918870937 -59.76746358 25.8312545
## [19,] -10.731529658 -35.948611804 -7.179126200 -66.85714696 28.5700146
## [20,] -10.307142820 -32.921053865 -7.054649841 -63.33134958 27.7040955
## [21,] -10.605441880 -35.358441017 -7.141036700 -65.30902736 27.8054156
## [22,] -10.601638528 -34.444627133 -7.060924703 -64.78701270 27.9108573
## [23,] -10.619776437 -35.935579438 -7.278560024 -65.23691699 27.4065078
## [24,] -10.749232186 -34.741556882 -6.855069628 -64.69574415 27.6451219
## [25,] -10.176969484 -34.860754934 -7.647543854 -65.00414181 27.9379705
## [26,] -11.778315627 -34.443656816 -5.356285443 -64.49561408 27.3658165
## [27,] -5.083014189 -21.885303756 -6.253987318 -35.90229819 14.3485243
## [28,] -21.157626537 -68.420486480 -14.444248317 -136.10893269 60.9067386
## [29,] -4.137042449 -10.586859782 -0.672665658 -17.84116589 6.7818590
## [30,] -0.070408125 -0.002670521 0.070414230 25.62031802 -24.5470106
## [31,] -2.341575908 -22.138748476 0.808483385 30.65528907 -44.6720887
## [32,] -0.538482071 -6.080005535 0.513058450 29.83931898 -34.2496419
## [33,] -2.396746992 -19.133105964 0.365214361 26.35911590 -39.5786817
## [34,] -1.521708760 -13.255033916 0.342475104 27.54819947 -36.8608269
## [35,] -1.846727809 -14.996284574 0.792736459 28.18118426 -38.5219507
## [36,] -1.545437700 -14.528844336 -0.069101549 27.27614210 -37.3659253
## [37,] -2.203703368 -14.709916487 1.248161597 27.74011901 -38.0464713
## [38,] 1.010145041 -12.169680475 -0.702054674 30.17505018 -38.2443581
## [39,] -3.405316281 -2.814494890 4.042749140 13.10173031 -16.6982062
## [40,] 0.411762089 -47.842678882 -13.307209684 62.15415284 -88.7573107
## [41,] 10.068105802 -6.333590093 -28.986649463 12.55307799 -6.9996032
## [42,] 5.014609818 13.921412627 3.772955046 31.85833668 -16.0767321
## [43,] 6.541099160 21.238443317 4.417969276 41.07456033 -16.6622891
## [44,] 5.653487685 16.252647406 4.116682277 36.74848663 -16.8177166
## [45,] 6.306293142 20.499552419 4.273640481 39.07063768 -15.9548683
## [46,] 6.097732384 18.597777274 4.196542917 38.34638496 -16.5886084
## [47,] 6.184556585 20.310123809 4.303925136 38.67197575 -15.7633975
## [48,] 6.352243608 19.505001434 4.120583140 38.37085097 -16.0734480
## [49,] 5.234501891 17.761771844 4.273011862 37.42437386 -16.2335361
## [50,] 8.440539264 21.577839477 3.835460721 39.82322697 -16.1506969
## [51,] 1.402540329 8.181551461 1.914216914 15.62266077 -5.4671268
## [52,] 7.399377066 44.264339084 25.161306738 86.49073876 -42.0093453
## [53,] -10.018446476 3.088368689 26.260980744 -10.00434450 1.8975473
## [54,] -0.896648322 -3.090650723 -0.686695934 -14.33297741 12.4856343
## [55,] -0.863264553 0.928888714 -1.284629639 -18.95233797 20.0048019
## [56,] -1.073848978 -2.242441260 -1.066917908 -17.65896008 16.8927968
## [57,] -0.774076907 0.318658138 -1.096621177 -17.24354182 18.1926227
## [58,] -0.715395931 0.124572804 -1.010806578 -16.69881366 17.6138523
## [59,] -1.062019971 -1.396538510 -1.239347483 -18.14878073 17.8606101
## [60,] -0.650305650 0.053900370 -0.885425865 -16.53206037 17.4380732
## [61,] -1.504353976 -1.992369248 -1.426641588 -19.15954813 18.2864862
## [62,] -2.264499205 -1.824272918 -0.909378614 -18.20703539 17.8000529
## [63,] 0.780978343 -0.616090353 -0.713770334 -6.14281601 6.3959035
## [64,] -0.105162870 1.576928492 -7.538508677 -41.40845344 45.2508728
## [65,] 3.962686619 2.117420499 -6.265636012 6.26894689 -0.1324686
## [66,] -0.005279742 -0.018512983 0.004822222 0.01513112 -0.0293585
## [,16] [,17] [,18] [,19] [,20]
## [1,] 9.90934717 -0.76035765 17.06220167 18.06305352 16.90037685
## [2,] -9.84670977 2.00858049 -16.64028877 -17.16362002 -16.93537351
## [3,] -9.77314875 0.11417643 -17.06249770 -18.43800514 -16.45467248
## [4,] -9.63506599 1.01422494 -16.47848453 -16.89478367 -17.28742691
## [5,] -10.10646735 0.61865817 -17.47520120 -18.67225376 -16.94148139
## [6,] -9.88754012 0.89303340 -16.98097321 -17.83201329 -17.11293808
## [7,] -9.89078740 0.66882835 -17.06271929 -18.21024143 -16.63858279
## [8,] -9.95780129 0.82538395 -17.12308339 -18.07433576 -17.07199770
## [9,] -9.93387184 0.74416811 -17.10830076 -18.15308254 -16.88182080
## [10,] -9.77922883 0.76096423 -16.84652180 -17.78700563 -16.73827767
## [11,] -6.10507865 0.87352623 -10.23786993 -10.73152966 -10.30714282
## [12,] -19.17751343 0.44424116 -33.68576103 -35.94861180 -32.92105386
## [13,] -4.21899076 1.09268443 -6.91887094 -7.17912620 -7.05464984
## [14,] -38.37243595 17.38655308 -59.76746358 -66.85714696 -63.33134958
## [15,] 16.23478456 -17.82269536 25.83125451 28.57001461 27.70409550
## [16,] 23.18907834 -9.69819839 35.21187906 39.41329313 37.69047521
## [17,] -9.69819839 9.52735581 -16.50328430 -17.97508600 -17.05031496
## [18,] 35.21187906 -16.50328430 56.53648888 60.89992118 58.86415185
## [19,] 39.41329313 -17.97508600 60.89992118 70.56789120 63.53981596
## [20,] 37.69047521 -17.05031496 58.86415185 63.53981596 65.35689152
## [21,] 38.55257779 -17.51125849 60.00313815 68.02507052 62.37332379
## [22,] 38.40367666 -17.38883102 59.85975974 66.18488315 64.47515598
## [23,] 38.50589157 -17.36631495 60.01712302 67.73095492 62.70139277
## [24,] 38.28509491 -17.32445911 59.64072372 66.60934762 63.32422798
## [25,] 38.46668473 -17.46148721 59.90455304 67.01823588 63.48447417
## [26,] 38.15438678 -17.20615019 59.44637229 66.45346400 63.02479044
## [27,] 21.18607324 -9.34590620 33.40729733 36.98409016 34.87314432
## [28,] 80.64405249 -37.24881551 124.62326971 140.53728633 133.07017415
## [29,] 10.58849868 -4.52024050 17.00552532 17.77797200 18.27351545
## [30,] -14.39099057 12.68778717 -23.01850979 -27.42060806 -23.60246231
## [31,] -18.70670720 20.03422469 -28.94351257 -29.04061721 -34.32634244
## [32,] -16.76288293 17.22241559 -27.26409674 -33.66133197 -23.67826627
## [33,] -15.69948431 18.04289103 -24.71245564 -26.08749004 -28.60136362
## [34,] -15.97203190 17.50479258 -25.45669127 -29.15442411 -25.84129686
## [35,] -16.55435272 18.03986460 -26.34907934 -28.45923757 -29.19987437
## [36,] -15.88880721 17.60744756 -25.31346805 -28.38393322 -26.57464227
## [37,] -16.26782870 17.82601346 -25.87057148 -28.29168164 -28.20193006
## [38,] -17.56857271 18.27804681 -27.98840201 -31.44216443 -29.29556496
## [39,] -7.18192008 8.31211896 -12.02214028 -13.63020980 -12.83186284
## [40,] -37.60059990 40.42918514 -58.32321303 -63.20467739 -63.39866002
## [41,] -6.62966813 4.66001115 -11.33653988 -13.46884355 -11.31227311
## [42,] -19.21610398 9.29496346 -30.15001043 -32.37629786 -31.72766254
## [43,] -24.77859061 10.08945529 -37.36511932 -42.65829286 -39.67366730
## [44,] -22.44056648 9.58674454 -33.96638129 -36.65466521 -38.41544807
## [45,] -23.50764183 9.74009336 -35.76400576 -40.56522616 -37.43945119
## [46,] -23.23089008 9.78835050 -35.27635529 -39.00638187 -38.46324330
## [47,] -23.28716783 9.61624540 -35.42701200 -40.05209642 -37.26118381
## [48,] -23.16207719 9.66212414 -35.18689154 -39.51435252 -37.46170324
## [49,] -22.70151157 9.53118820 -34.41072606 -38.10464183 -37.45587687
## [50,] -23.96099839 9.89693734 -36.49881388 -41.19973303 -38.50048932
## [51,] -9.55632821 3.36789876 -14.42842788 -15.92891176 -15.48293188
## [52,] -52.27325321 24.01020822 -79.25613704 -88.30706867 -86.30179515
## [53,] 5.48075774 -2.21487535 8.91269467 11.60257818 7.31881404
## [54,] 7.82505627 -7.04950919 13.22904979 15.13114490 13.51717029
## [55,] 10.79796230 -10.39806147 18.11665431 18.90775982 19.65631609
## [56,] 9.68261285 -9.33549131 16.57029992 19.20429186 15.39458018
## [57,] 9.67606187 -9.60313145 16.43940460 17.50640916 17.58541663
## [58,] 9.29612984 -9.36227386 15.79635096 17.50137664 15.93697859
## [59,] 10.12011358 -9.65513665 17.25448716 18.64726000 17.97954933
## [60,] 9.18145516 -9.30080060 15.69084274 17.18035313 16.02797748
## [61,] 10.72680782 -9.90612369 18.13724569 19.85843246 18.68604319
## [62,] 10.13315131 -9.65012532 17.22833125 18.99943350 17.48511032
## [63,] 3.18970141 -3.60033825 5.77654151 6.44613581 5.86605061
## [64,] 23.66963397 -23.35732514 39.60158092 42.15618815 41.88688202
## [65,] -3.98133925 0.44505280 -5.70642567 -6.91622457 -5.16120685
## [66,] -0.00930836 0.01249415 -0.01403533 -0.01474488 -0.01684455
## [,21] [,22] [,23] [,24] [,25]
## [1,] 17.80331477 17.51414120 18.02361343 17.61123333 17.58083482
## [2,] -17.04464297 -17.12125305 -17.13198752 -17.04504583 -17.06412622
## [3,] -18.06391259 -17.42503081 -18.36993574 -17.69105478 -17.62546414
## [4,] -16.68962629 -17.23389431 -16.83246531 -16.89183688 -16.92570197
## [5,] -18.41958706 -17.70430506 -18.81911933 -18.11768202 -17.96155655
## [6,] -17.55021435 -17.69557366 -17.60584731 -17.28663437 -17.74846540
## [7,] -17.97404899 -17.27300546 -18.31928430 -18.61991637 -16.44011338
## [8,] -17.80769332 -17.60605072 -18.05417993 -16.33725125 -19.16443924
## [9,] -17.89671207 -17.57133033 -18.07844094 -18.46270821 -16.64388335
## [10,] -17.53293533 -17.19054894 -17.90147620 -16.92848245 -18.18149725
## [11,] -10.60544188 -10.60163853 -10.61977644 -10.74923219 -10.17696948
## [12,] -35.35844102 -34.44462713 -35.93557944 -34.74155688 -34.86075493
## [13,] -7.14103670 -7.06092470 -7.27856002 -6.85506963 -7.64754385
## [14,] -65.30902736 -64.78701270 -65.23691699 -64.69574415 -65.00414181
## [15,] 27.80541557 27.91085727 27.40650782 27.64512186 27.93797048
## [16,] 38.55257779 38.40367666 38.50589157 38.28509491 38.46668473
## [17,] -17.51125849 -17.38883102 -17.36631495 -17.32445912 -17.46148721
## [18,] 60.00313815 59.85975974 60.01712302 59.64072372 59.90455304
## [19,] 68.02507052 66.18488315 67.73095492 66.60934762 67.01823588
## [20,] 62.37332379 64.47515598 62.70139277 63.32422798 63.48447417
## [21,] 66.76365026 64.31789877 66.36327698 65.02940000 65.46681065
## [22,] 64.31789877 66.95415513 63.53556681 65.10576558 64.60171617
## [23,] 66.36327698 63.53556681 67.63114463 64.52620922 65.70231887
## [24,] 65.02940000 65.10576558 64.52620922 68.24194378 60.36474820
## [25,] 65.46681065 64.60171617 65.70231887 60.36474820 71.79176265
## [26,] 64.89840500 64.85431318 64.40209286 67.65983986 59.02170826
## [27,] 36.29380061 35.68065626 36.35803723 33.87075318 39.06672101
## [28,] 136.92949107 135.90095304 136.90675186 136.17398487 136.05002512
## [29,] 17.46825193 17.98416298 17.82572495 19.87801123 15.04635062
## [30,] -26.40973541 -25.16731374 -26.33168879 -25.47540360 -25.70333893
## [31,] -29.16148868 -32.14738611 -28.78237086 -30.69262052 -30.88226944
## [32,] -32.43626054 -27.35335855 -31.95694373 -29.49671059 -29.91055076
## [33,] -25.33052218 -27.96924434 -24.51163628 -26.33658579 -26.62661897
## [34,] -28.47831597 -25.65124649 -28.88436448 -27.37050535 -27.58140234
## [35,] -27.60721119 -29.77434288 -26.88025357 -26.95402658 -29.79843948
## [36,] -27.69471540 -26.83012391 -27.57104366 -30.88850275 -22.86248540
## [37,] -27.46256651 -28.16983427 -26.96560455 -23.16013834 -33.06743900
## [38,] -30.67420353 -30.58677165 -29.48003092 -31.25710035 -27.77096344
## [39,] -13.16986263 -12.88569926 -13.44091217 -12.87804592 -13.72618413
## [40,] -61.68377536 -63.06401520 -60.35782931 -61.80987617 -62.41693100
## [41,] -12.99594023 -12.65917596 -12.31740712 -12.40861723 -11.86724851
## [42,] -31.85282637 -31.99719321 -31.75825872 -31.79354014 -31.95172981
## [43,] -41.52714946 -40.87709389 -41.44129983 -40.95020817 -41.17750887
## [44,] -36.00672270 -37.52987274 -36.09344768 -36.78397545 -36.81825244
## [45,] -39.77082566 -38.58281673 -39.70219426 -38.87958239 -39.21537210
## [46,] -38.00334353 -40.13629810 -37.10698666 -38.78100108 -37.93345276
## [47,] -39.25861650 -37.37919636 -40.26877092 -37.77848513 -39.61217103
## [48,] -38.69048171 -38.59341733 -38.44356721 -43.69283565 -31.47075289
## [49,] -37.14386362 -37.27181556 -37.38615777 -27.98832763 -50.63673109
## [50,] -40.37055637 -40.49126744 -39.38170305 -45.20340700 -30.43270779
## [51,] -15.61357752 -15.29933490 -15.93520451 -11.70271202 -22.01499834
## [52,] -86.11827019 -86.31861444 -87.13466632 -87.36493825 -86.40825190
## [53,] 11.44912228 10.09904948 9.64084245 5.39726374 15.02664978
## [54,] 14.64433369 14.16627605 14.58889956 14.26811511 14.37755040
## [55,] 18.64318544 19.36726906 18.51098968 18.93456854 19.03948145
## [56,] 18.59821537 16.71531820 18.40289474 17.51609880 17.71306545
## [57,] 17.03092336 17.79182045 16.76295745 17.23212924 17.31902664
## [58,] 17.11107772 15.52970328 17.48587162 16.43748560 16.93895550
## [59,] 18.11436480 18.92979575 17.76060033 17.77148216 18.62402530
## [60,] 16.77022501 16.25276897 16.87585446 19.64085210 12.68457350
## [61,] 19.32621772 19.33002052 18.75376887 14.30616375 24.99970665
## [62,] 18.55253343 18.50987137 17.90545181 20.67036742 13.86224634
## [63,] 6.23915780 5.92038559 6.36998799 4.39538166 8.97469823
## [64,] 41.05773107 41.48777587 41.21344736 41.74528764 41.51261134
## [65,] -6.92653000 -6.39025396 -6.04203296 -4.24864848 -8.42479385
## [66,] -0.01430868 -0.01546855 -0.01461416 -0.01567292 -0.01509746
## [,26] [,27] [,28] [,29] [,30]
## [1,] 17.66178815 10.553316210 34.93153964 5.63622874 0.060843800
## [2,] -17.03299336 -9.977748804 -34.38632113 -5.49563391 -0.848096827
## [3,] -17.78361907 -10.745422249 -34.75744987 -5.57122380 0.368597037
## [4,] -16.90956255 -10.027585780 -33.70037443 -5.58185142 -0.992421705
## [5,] -18.17838954 -10.865085224 -35.74475090 -5.73948627 0.441222068
## [6,] -17.35785509 -10.561538951 -34.74377255 -5.49253627 -0.248511075
## [7,] -18.53912082 -10.044115868 -35.16839947 -6.40123062 0.086034082
## [8,] -16.62418254 -11.320383406 -34.72374011 -4.53689579 -0.086788667
## [9,] -18.49176181 -10.258619958 -35.03654249 -5.88253082 -0.011087859
## [10,] -15.20933589 -10.973416695 -35.21641399 -6.46356735 -0.170253016
## [11,] -11.77831563 -5.083014189 -21.15762654 -4.13704245 -0.070408125
## [12,] -34.44365682 -21.885303756 -68.42048648 -10.58685978 -0.002670521
## [13,] -5.35628544 -6.253987318 -14.44424832 -0.67266566 0.070414230
## [14,] -64.49561408 -35.902298193 -136.10893269 -17.84116589 25.620318019
## [15,] 27.36581648 14.348524341 60.90673860 6.78185902 -24.547010583
## [16,] 38.15438678 21.186073239 80.64405249 10.58849868 -14.390990568
## [17,] -17.20615019 -9.345906204 -37.24881552 -4.52024050 12.687787166
## [18,] 59.44637229 33.407297327 124.62326971 17.00552532 -23.018509785
## [19,] 66.45346400 36.984090160 140.53728633 17.77797200 -27.420608063
## [20,] 63.02479044 34.873144321 133.07017415 18.27351545 -23.602462306
## [21,] 64.89840500 36.293800607 136.92949107 17.46825193 -26.409735414
## [22,] 64.85431318 35.680656259 135.90095304 17.98416298 -25.167313743
## [23,] 64.40209286 36.358037230 136.90675186 17.82572495 -26.331688786
## [24,] 67.65983986 33.870753183 136.17398487 19.87801123 -25.475403600
## [25,] 59.02170826 39.066721007 136.05002512 15.04635062 -25.703338935
## [26,] 74.69877594 29.621938546 135.98116099 23.17144040 -25.412992474
## [27,] 29.62193855 30.672341605 69.99708674 -5.66082913 -13.934494810
## [28,] 135.98116099 69.997086740 294.12184323 52.71604992 -54.679719445
## [29,] 23.17144040 -5.660829129 52.71604992 43.43433194 -5.874538583
## [30,] -25.41299247 -13.934494810 -54.67971945 -5.87453858 27.435195518
## [31,] -30.19299554 -15.321041970 -68.12028897 -8.63029219 22.489583841
## [32,] -29.39982551 -16.165008927 -64.17043087 -6.12025683 29.451550502
## [33,] -25.97446431 -13.212920011 -58.56535642 -7.07972581 22.381479212
## [34,] -27.07349910 -14.376752893 -60.29856611 -6.40967453 25.053119582
## [35,] -26.93542530 -15.132827288 -61.56958074 -6.19940357 24.433899938
## [36,] -29.34052557 -11.378719661 -61.98383587 -12.13908573 24.622820629
## [37,] -25.82813159 -15.911927107 -59.92477940 -4.09885627 24.280531010
## [38,] -37.12454117 -16.896143151 -57.50315840 7.65745138 26.023054323
## [39,] -10.01547941 -9.670398091 -31.51141197 -6.42735005 14.059639407
## [40,] -62.22084388 -31.707870241 -133.40178807 -8.89271198 48.649593943
## [41,] -16.33018863 -5.151855532 -17.40125707 10.41553423 11.221204914
## [42,] -31.64240570 -17.596796291 -66.91010467 -9.07965951 10.417643337
## [43,] -40.83591704 -22.702607048 -86.32086970 -11.13215398 16.165790163
## [44,] -36.55739667 -20.042528209 -77.59288220 -10.75026167 12.303873375
## [45,] -38.80025336 -21.713063768 -81.92942882 -10.48639325 15.301696166
## [46,] -38.66477091 -21.064277623 -80.36281336 -10.26648727 14.061285773
## [47,] -37.64513880 -21.576892720 -81.50686962 -11.05118884 14.972346101
## [48,] -43.12918848 -18.652320288 -80.71433747 -12.51736003 14.498618930
## [49,] -27.03033349 -23.538727097 -81.79748900 -13.60832860 13.637437702
## [50,] -58.51989630 -16.147884623 -77.38533429 -4.55176510 15.397177041
## [51,] -2.61317826 -21.826311040 -30.26276037 8.80626515 4.935636611
## [52,] -84.55102387 -32.580141047 -211.34257776 -82.85952949 33.919579251
## [53,] 3.25538851 40.116813606 -29.74067996 -107.80247223 -5.719513426
## [54,] 14.21763761 7.838274940 30.42825188 3.56858411 -12.181178106
## [55,] 18.76093219 10.061189498 40.79808615 5.21655874 -12.564400882
## [56,] 17.42415163 9.633534390 37.63654240 4.33759582 -13.862482795
## [57,] 17.10979946 9.202552702 37.01886729 4.58486160 -12.289285567
## [58,] 16.25369501 8.926749649 36.07102720 4.64151573 -12.286602764
## [59,] 17.88170623 10.203493299 38.26136752 3.62628210 -13.233171515
## [60,] 18.13801297 5.715229155 38.64626988 11.90769142 -12.374036210
## [61,] 17.66804616 16.047903001 33.95644894 -10.85481366 -13.522390129
## [62,] 27.30207945 6.885576344 35.64476051 1.23023729 -13.280353336
## [63,] 0.05255964 10.025298941 11.60370991 -5.66569796 -5.837799736
## [64,] 40.15133887 14.539993120 103.25504640 40.06753197 -27.551360776
## [65,] -3.33178523 -20.975360846 11.93592559 53.01062448 0.532101724
## [66,] -0.01408504 0.002209006 -0.05057119 -0.03768717 0.015376186
## [,31] [,32] [,33] [,34] [,35]
## [1,] 8.03471684 2.16466871 7.20425838 4.93159528 5.56376923
## [2,] -1.01602835 -0.64628252 -1.76839285 -1.27958395 -1.16338863
## [3,] -12.47647017 -2.29441716 -10.40797502 -6.70572910 -8.02276622
## [4,] -4.59097889 -4.35143974 -4.42689673 -4.75780405 -3.67755348
## [5,] -9.80347932 -1.27075065 -9.11763426 -4.73224151 -6.80523620
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## [13,] 0.80848339 0.51305845 0.36521436 0.34247510 0.79273646
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## [30,] 22.48958384 29.45155050 22.38147921 25.05311958 24.43389994
## [31,] 62.16679848 29.63235528 50.79935123 41.12393841 46.68233858
## [32,] 29.63235528 48.83387104 28.93357331 38.53223339 31.42583622
## [33,] 50.79935123 28.93357331 45.83935895 34.76963648 42.53046388
## [34,] 41.12393841 38.53223339 34.76963648 41.27890092 33.66635924
## [35,] 46.68233858 31.42583622 42.53046388 33.66635924 43.33916327
## [36,] 43.12570994 35.67221709 37.67123195 38.25990781 34.49488601
## [37,] 45.48966649 32.95835473 40.61061108 35.71596152 41.35451094
## [38,] 43.60409880 36.27344769 38.90170109 37.59001656 38.52225340
## [39,] 18.16697716 16.55581124 16.89893584 16.35976697 16.94636965
## [40,] 109.48841503 74.97315811 94.80850603 85.46481819 90.24929142
## [41,] 4.42289499 10.71270258 5.10231437 7.76496518 6.26668502
## [42,] 19.67471271 14.88970663 16.37958891 15.58350880 16.45253972
## [43,] 18.29719954 18.72480770 15.51524317 16.74092535 16.78624738
## [44,] 22.79035489 12.09863889 18.31723844 15.52726888 17.71004477
## [45,] 16.91976090 18.64814363 14.36203654 16.28575137 15.93368510
## [46,] 20.15563531 14.93033039 17.42623295 14.39282303 18.19147048
## [47,] 16.93486379 18.41161864 13.83566161 17.09661115 15.32825917
## [48,] 18.19458225 17.23410192 15.12728123 16.33140807 14.24323757
## [49,] 19.73429140 15.12973965 16.66877447 15.02454076 19.96696112
## [50,] 17.43731772 17.94918439 14.91064508 15.98395765 15.32223947
## [51,] 6.56926854 5.60204562 5.27069570 5.46181157 6.43572779
## [52,] 50.93205378 39.79060868 42.73331689 40.62205256 42.47366761
## [53,] 2.54638401 -7.48225739 1.31528250 -3.21690867 -2.98953411
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## [65,] -1.03246120 1.92395552 -1.05197250 0.55907492 0.89062663
## [66,] 0.03826785 0.02197809 0.03296227 0.02746149 0.02996011
## [,36] [,37] [,38] [,39] [,40]
## [1,] 5.35973243 5.54308261 4.363227292 1.86959790 14.39762841
## [2,] -1.39305210 -1.24279819 -0.582645448 -0.97138757 -2.08933815
## [3,] -7.40930013 -7.89532758 -6.320637158 -2.37490859 -20.98970934
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## [5,] -5.70727984 -6.48015276 -4.889216144 -1.96746077 -16.67132173
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## [8,] -7.45481633 -3.42085953 -4.183148594 -1.89143480 -14.19993422
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## [13,] -0.06910155 1.24816160 -0.702054674 4.04274914 -13.30720968
## [14,] 27.27614210 27.74011901 30.175050178 13.10173031 62.15415284
## [15,] -37.36592529 -38.04647131 -38.244358116 -16.69820624 -88.75731067
## [16,] -15.88880721 -16.26782870 -17.568572714 -7.18192008 -37.60059990
## [17,] 17.60744756 17.82601346 18.278046814 8.31211896 40.42918514
## [18,] -25.31346805 -25.87057148 -27.988402008 -12.02214028 -58.32321303
## [19,] -28.38393322 -28.29168164 -31.442164429 -13.63020979 -63.20467739
## [20,] -26.57464227 -28.20193006 -29.295564962 -12.83186284 -63.39866002
## [21,] -27.69471540 -27.46256651 -30.674203528 -13.16986263 -61.68377535
## [22,] -26.83012391 -28.16983427 -30.586771652 -12.88569926 -63.06401520
## [23,] -27.57104366 -26.96560455 -29.480030921 -13.44091217 -60.35782931
## [24,] -30.88850275 -23.16013834 -31.257100352 -12.87804592 -61.80987617
## [25,] -22.86248540 -33.06743900 -27.770963444 -13.72618413 -62.41693100
## [26,] -29.34052557 -25.82813159 -37.124541173 -10.01547941 -62.22084388
## [27,] -11.37871966 -15.91192711 -16.896143150 -9.67039809 -31.70787024
## [28,] -61.98383587 -59.92477940 -57.503158398 -31.51141197 -133.40178806
## [29,] -12.13908573 -4.09885627 7.657451380 -6.42735005 -8.89271198
## [30,] 24.62282063 24.28053101 26.023054323 14.05963941 48.64959394
## [31,] 43.12570994 45.48966649 43.604098800 18.16697716 109.48841503
## [32,] 35.67221709 32.95835473 36.273447686 16.55581124 74.97315811
## [33,] 37.67123195 40.61061108 38.901701095 16.89893584 94.80850603
## [34,] 38.25990781 35.71596152 37.590016560 16.35976697 85.46481819
## [35,] 34.49488601 41.35451094 38.522253398 16.94636965 90.24929142
## [36,] 44.70282163 30.21654985 34.939680362 17.93303979 86.80509787
## [37,] 30.21654985 53.46463852 37.079425170 17.94145253 88.95305539
## [38,] 34.93968036 37.07942517 72.739550438 0.10693023 89.57917630
## [39,] 17.93303979 17.94145253 0.106930230 22.34840335 28.40118315
## [40,] 86.80509787 88.95305539 89.579176299 28.40118315 258.07233585
## [41,] 5.67885870 5.35642175 21.325373411 -14.71128528 81.44604186
## [42,] 15.67939365 16.16909819 16.942290845 6.26968488 39.50252998
## [43,] 16.42743341 16.60225997 18.256840530 7.66471743 37.71098191
## [44,] 16.08483019 17.14467772 17.425024491 7.00859620 40.64535131
## [45,] 15.81125248 15.85258266 17.603594889 7.25249427 36.19050859
## [46,] 15.91024376 16.52880930 18.742687546 6.85827796 38.93756797
## [47,] 15.41981797 16.25585812 15.959813116 7.67783639 35.72696826
## [48,] 22.07494519 8.26407460 20.368091108 6.62877068 36.45158137
## [49,] 6.45709926 31.15142830 3.901874407 11.49674099 38.46242694
## [50,] 16.73209011 14.84637845 51.168839942 -6.00300966 33.98497455
## [51,] 2.78427593 7.01738882 -8.509582665 10.83191002 20.20305797
## [52,] 49.39016484 39.77581848 3.397067174 42.69021113 45.37736097
## [53,] 13.88813691 -6.87803140 -62.877547093 33.82436811 -97.08399657
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## [58,] -18.06090610 -17.33714850 -17.488523021 -8.34901197 -40.05200990
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## [61,] -9.34164576 -29.76445980 -32.605243914 -2.65201397 -43.71316564
## [62,] -17.67492587 -16.88511454 -38.084016874 0.18321138 -37.78649364
## [63,] -5.24222638 -7.36366916 1.398571476 -9.14093512 -15.96200479
## [64,] -48.84247717 -44.12224088 -23.363052410 -31.04289719 -90.13795065
## [65,] -8.05601492 3.01203601 33.190702705 -16.86347636 33.71819053
## [66,] 0.03365210 0.02844412 0.003357603 0.02499368 0.05096532
## [,41] [,42] [,43] [,44] [,45]
## [1,] -2.87192569 -7.61577417 -10.814243986 -8.75297809 -10.431792541
## [2,] 1.49957948 8.62797251 10.349967628 9.50826202 10.003006379
## [3,] 3.63382737 6.99019434 10.878533532 7.98912305 10.573715188
## [4,] 1.99371167 7.82187504 10.229020981 9.48226989 9.738729826
## [5,] 3.41278910 7.53596950 11.154180570 8.49766975 10.862191832
## [6,] 2.48350564 7.73011390 10.703403141 8.99360135 10.272885212
## [7,] 3.44277844 7.50904774 10.865701326 8.51464154 10.529351827
## [8,] 2.24266129 7.69702376 10.837176905 8.88481913 10.444165528
## [9,] 5.15444444 7.61119543 10.857822762 8.72209916 10.478794166
## [10,] -2.86598201 7.54945222 10.655336073 8.70254305 10.271923858
## [11,] 10.06810580 5.01460982 6.541099160 5.65348768 6.306293142
## [12,] -6.33359009 13.92141263 21.238443316 16.25264741 20.499552419
## [13,] -28.98664946 3.77295505 4.417969276 4.11668228 4.273640481
## [14,] 12.55307799 31.85833668 41.074560325 36.74848663 39.070637684
## [15,] -6.99960316 -16.07673212 -16.662289102 -16.81771664 -15.954868259
## [16,] -6.62966813 -19.21610398 -24.778590611 -22.44056648 -23.507641832
## [17,] 4.66001115 9.29496346 10.089455289 9.58674454 9.740093363
## [18,] -11.33653988 -30.15001043 -37.365119323 -33.96638129 -35.764005760
## [19,] -13.46884355 -32.37629786 -42.658292863 -36.65466521 -40.565226162
## [20,] -11.31227311 -31.72766254 -39.673667301 -38.41544807 -37.439451187
## [21,] -12.99594023 -31.85282637 -41.527149461 -36.00672270 -39.770825664
## [22,] -12.65917596 -31.99719321 -40.877093887 -37.52987274 -38.582816735
## [23,] -12.31740712 -31.75825872 -41.441299835 -36.09344768 -39.702194260
## [24,] -12.40861723 -31.79354014 -40.950208171 -36.78397545 -38.879582390
## [25,] -11.86724851 -31.95172981 -41.177508868 -36.81825244 -39.215372097
## [26,] -16.33018863 -31.64240570 -40.835917042 -36.55739667 -38.800253359
## [27,] -5.15185553 -17.59679629 -22.702607048 -20.04252821 -21.713063768
## [28,] -17.40125707 -66.91010467 -86.320869702 -77.59288220 -81.929428824
## [29,] 10.41553423 -9.07965951 -11.132153982 -10.75026167 -10.486393255
## [30,] 11.22120491 10.41764334 16.165790163 12.30387337 15.301696166
## [31,] 4.42289499 19.67471271 18.297199540 22.79035489 16.919760905
## [32,] 10.71270258 14.88970663 18.724807696 12.09863889 18.648143626
## [33,] 5.10231437 16.37958891 15.515243170 18.31723844 14.362036541
## [34,] 7.76496518 15.58350880 16.740925350 15.52726888 16.285751365
## [35,] 6.26668502 16.45253972 16.786247385 17.71004477 15.933685098
## [36,] 5.67885870 15.67939365 16.427433405 16.08483019 15.811252480
## [37,] 5.35642175 16.16909819 16.602259967 17.14467772 15.852582655
## [38,] 21.32537341 16.94229085 18.256840530 17.42502449 17.603594890
## [39,] -14.71128528 6.26968488 7.664717431 7.00859620 7.252494272
## [40,] 81.44604187 39.50252998 37.710981907 40.64535131 36.190508593
## [41,] 143.87797531 4.34537149 7.595755554 5.20267164 7.243447596
## [42,] 4.34537149 17.51161339 20.022597599 19.16834549 19.229383930
## [43,] 7.59575555 20.02259760 26.769806125 23.43672031 25.340354110
## [44,] 5.20267164 19.16834549 23.436720306 23.66425720 21.932717218
## [45,] 7.24344760 19.22938393 25.340354110 21.93271722 24.253887277
## [46,] 6.80100191 19.38251114 24.642589155 23.05271215 23.146342617
## [47,] 6.46055748 19.08775832 25.069783644 21.89052058 24.020003032
## [48,] 6.05239423 19.14595464 24.788919663 22.29303589 23.502874848
## [49,] 3.62622797 18.95656541 24.130663554 22.42090769 22.849267647
## [50,] 12.03271289 19.63371049 25.733597610 22.64422547 24.483688379
## [51,] 8.75236154 7.91598023 10.192425907 9.31673877 9.693314787
## [52,] -92.98293173 43.96938755 55.521036817 51.93998928 52.365595044
## [53,] -191.41332815 -3.69222738 -6.450258832 -3.02987443 -6.620476406
## [54,] -5.29832316 -6.38308498 -8.539266817 -7.04530839 -8.163185231
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## [57,] -4.33828599 -9.42411585 -9.909898177 -10.06612478 -9.457736709
## [58,] -4.25579278 -9.02155689 -9.730458662 -9.11760535 -9.432003909
## [59,] -5.38016426 -9.52597304 -10.532020476 -9.88398954 -10.184947529
## [60,] -0.61256812 -8.86435149 -9.568612788 -9.03618100 -9.244604386
## [61,] -13.54872197 -10.05797987 -11.226757987 -10.36914758 -10.890231553
## [62,] -7.35639162 -9.53051373 -10.632591828 -9.68357091 -10.303942406
## [63,] -4.22522925 -2.77182560 -3.437563280 -2.93011223 -3.318016365
## [64,] 29.97960376 -23.91665029 -24.088088747 -24.76985729 -23.149958374
## [65,] 66.43530900 3.25250821 4.380629757 3.04513415 4.425795757
## [66,] -0.04157824 0.01033157 0.008997963 0.01144503 0.008324022
## [,46] [,47] [,48] [,49] [,50]
## [1,] -9.703240201 -10.344041265 -10.018976568 -9.38724293 -10.73455470
## [2,] 9.851933770 9.943968854 9.856687034 9.59190963 10.30103138
## [3,] 9.383388977 10.470935370 9.965600415 9.03982573 10.85881561
## [4,] 9.840229917 9.630485203 9.582264479 9.47749710 10.17048367
## [5,] 9.593810043 10.878299199 10.372929858 9.28934986 11.12118330
## [6,] 9.974975619 10.046678872 9.585866449 10.04228922 10.38332549
## [7,] 9.462673694 10.451442528 11.541778526 6.91720245 11.88421230
## [8,] 9.745567719 10.506599633 8.053162176 12.53849870 9.44786014
## [9,] 9.771030368 10.257279106 11.303350306 6.96696012 11.90057852
## [10,] 9.438593807 10.408699494 9.199919506 11.45790957 5.27823975
## [11,] 6.097732384 6.184556585 6.352243608 5.23450189 8.44053926
## [12,] 18.597777274 20.310123809 19.505001433 17.76177184 21.57783948
## [13,] 4.196542917 4.303925136 4.120583140 4.27301186 3.83546072
## [14,] 38.346384963 38.671975751 38.370850972 37.42437386 39.82322697
## [15,] -16.588608443 -15.763397521 -16.073447960 -16.23353606 -16.15069694
## [16,] -23.230890076 -23.287167833 -23.162077186 -22.70151157 -23.96099839
## [17,] 9.788350501 9.616245400 9.662124138 9.53118820 9.89693734
## [18,] -35.276355289 -35.427012002 -35.186891536 -34.41072606 -36.49881388
## [19,] -39.006381872 -40.052096425 -39.514352517 -38.10464183 -41.19973303
## [20,] -38.463243305 -37.261183807 -37.461703239 -37.45587687 -38.50048932
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## [63,] 3.73407652 3.54257625 3.67582262 3.68728486 2.76609613
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## [,61] [,62] [,63] [,64] [,65]
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## [25,] 24.999706654 13.862246337 8.974698229 41.51261134 -8.42479385
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## [27,] 16.047903001 6.885576344 10.025298941 14.53999312 -20.97536085
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## [32,] -17.757840154 -17.974510748 -6.903215814 -40.02617965 1.92395552
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## [34,] -17.402313470 -17.587292066 -6.439171365 -43.74039611 0.55907492
## [35,] -20.764096803 -17.469031929 -6.810370219 -45.73811714 0.89062663
## [36,] -9.341645762 -17.674925869 -5.242226384 -48.84247717 -8.05601492
## [37,] -29.764459802 -16.885114544 -7.363669162 -44.12224088 3.01203601
## [38,] -32.605243914 -38.084016874 1.398571476 -23.36305241 33.19070271
## [39,] -2.652013969 0.183211382 -9.140935116 -31.04289719 -16.86347636
## [40,] -43.713165639 -37.786493637 -15.962004787 -90.13795065 33.71819053
## [41,] -13.548721974 -7.356391618 -4.225229248 29.97960376 66.43530900
## [42,] -10.057979865 -9.530513730 -2.771825596 -23.91665029 3.25250821
## [43,] -11.226757986 -10.632591828 -3.437563279 -24.08808875 4.38062976
## [44,] -10.369147576 -9.683570912 -2.930112229 -24.76985729 3.04513415
## [45,] -10.890231552 -10.303942406 -3.318016365 -23.14995837 4.42579576
## [46,] -10.927029730 -10.904544029 -2.874991943 -23.47736109 4.85290218
## [47,] -10.766676593 -9.079590935 -3.681572265 -23.96406516 2.89523955
## [48,] -3.085776297 -14.694870707 -0.364810255 -23.08892734 2.68047793
## [49,] -20.386634212 2.933942152 -9.220427828 -31.84292055 -5.94825989
## [50,] -20.189434963 -38.688348531 11.241063272 -5.58951829 24.87242055
## [51,] -4.724997936 12.798548720 -15.747638623 -7.75288686 13.06448278
## [52,] 3.801602460 -8.279264875 -4.641868478 -130.35431757 -114.56834584
## [53,] 54.425213461 19.264956227 16.976740354 -117.29624144 -218.62095732
## [54,] 7.458746141 7.301683898 3.073738095 15.88062732 -0.34252337
## [55,] 10.746130731 10.370015373 3.725434425 26.37637189 -0.24732806
## [56,] 9.600300520 9.572318924 3.734076522 22.05940521 -0.71119123
## [57,] 10.071682605 9.704307683 3.542576252 23.84615763 -0.32049650
## [58,] 9.359745156 9.043867762 3.675822617 23.49104562 0.29598294
## [59,] 11.394189145 10.236143414 3.687284864 22.41614535 -2.35312164
## [60,] 0.419877525 7.851918225 2.766096128 29.74673954 12.01643547
## [61,] 28.856487627 15.679540578 4.131275654 8.70453479 -27.18055237
## [62,] 15.679540578 25.213130799 -4.395961998 13.43567727 -11.89216295
## [63,] 4.131275654 -4.395961998 9.158757080 7.71406441 -5.72098084
## [64,] 8.704534786 13.435677267 7.714064412 92.75382044 55.10536190
## [65,] -27.180552368 -11.892162950 -5.720980841 55.10536190 101.79573889
## [66,] 0.005026522 -0.001164111 -0.002851076 -0.07115196 -0.06573649
## [,66]
## [1,] 0.0083629520
## [2,] -0.0036912960
## [3,] -0.0110243865
## [4,] -0.0064235420
## [5,] -0.0094405203
## [6,] -0.0077187226
## [7,] -0.0084653387
## [8,] -0.0087450000
## [9,] -0.0090373689
## [10,] -0.0048693702
## [11,] -0.0052797415
## [12,] -0.0185129830
## [13,] 0.0048222221
## [14,] 0.0151311160
## [15,] -0.0293585011
## [16,] -0.0093083596
## [17,] 0.0124941464
## [18,] -0.0140353307
## [19,] -0.0147448825
## [20,] -0.0168445534
## [21,] -0.0143086780
## [22,] -0.0154685527
## [23,] -0.0146141579
## [24,] -0.0156729167
## [25,] -0.0150974647
## [26,] -0.0140850427
## [27,] 0.0022090058
## [28,] -0.0505711941
## [29,] -0.0376871689
## [30,] 0.0153761865
## [31,] 0.0382678509
## [32,] 0.0219780855
## [33,] 0.0329622673
## [34,] 0.0274614874
## [35,] 0.0299601132
## [36,] 0.0336520993
## [37,] 0.0284441242
## [38,] 0.0033576025
## [39,] 0.0249936847
## [40,] 0.0509653152
## [41,] -0.0415782443
## [42,] 0.0103315667
## [43,] 0.0089979631
## [44,] 0.0114450272
## [45,] 0.0083240224
## [46,] 0.0092210831
## [47,] 0.0094303426
## [48,] 0.0087353181
## [49,] 0.0190192928
## [50,] -0.0112765326
## [51,] 0.0005152604
## [52,] 0.1073712613
## [53,] 0.1410330161
## [54,] -0.0077045249
## [55,] -0.0150106984
## [56,] -0.0107408290
## [57,] -0.0131980859
## [58,] -0.0128443826
## [59,] -0.0111596882
## [60,] -0.0202256793
## [61,] 0.0050265216
## [62,] -0.0011641106
## [63,] -0.0028510761
## [64,] -0.0711519636
## [65,] -0.0657364872
## [66,] 0.0050831669
##
## $log_evidence
## [1] -367.8151
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 225
A function is defined for you in the code chunk below. This function
creates a coefficient summary plot in the style of the
coefplot() function, but uses the Bayesian results from the
Laplace Approximation. The first argument is the vector of posterior
means, and the second argument is the vector of posterior standard
deviations. The third argument is the name of the feature associated
with each coefficient.
viz_post_coefs <- function(post_means, post_sds, xnames)
{
tibble::tibble(
mu = post_means,
sd = post_sds,
x = xnames
) %>%
mutate(x = factor(x, levels = xnames)) %>%
ggplot(mapping = aes(x = x)) +
geom_hline(yintercept = 0, color = 'grey', linetype = 'dashed') +
geom_point(mapping = aes(y = mu)) +
geom_linerange(mapping = aes(ymin = mu - 2 * sd,
ymax = mu + 2 * sd,
group = x)) +
labs(x = 'feature', y = 'coefficient value') +
coord_flip() +
theme_bw()
}
Create the posterior summary visualization figure for model 3
and model 6. You must provide the posterior means and standard
deviations of the regression coefficients (the \(\beta\) parameters). Do NOT include the
\(\varphi\) parameter. The feature
names associated with the coefficients can be extracted from the design
matrix using the colnames() function.
### make the posterior coefficient visualization for model 3
post_means_3 <- laplace_03_weak$mode[1:ncol(X03)]
post_sds_3 <- sqrt(diag(laplace_03_weak$var_matrix)[1:ncol(X03)])
xnames_3 <- colnames(X03)
viz_post_coefs(post_means_3, post_sds_3, xnames_3)
### make the posterior coefficient visualization for model 6
post_means_6 <- laplace_06_weak$mode[1:ncol(X06)]
post_sds_6 <- sqrt(diag(laplace_06_weak$var_matrix)[1:ncol(X06)])
xnames_6 <- colnames(X06)
viz_post_coefs(post_means_6, post_sds_6, xnames_6)
Use the Bayes Factor to identify the better of the models.
bf <- exp(laplace_03_weak$log_evidence - laplace_06_weak$log_evidence)
bf
## [1] 1.483395e+88
bf is greater than 1, then Model 3 is favored over Model 6.
You fit the Bayesian models assuming a diffuse or weak prior. Let’s now try a more informative or strong prior by reducing the prior standard deviation on the regression coefficients from 50 to 1. The prior mean will still be zero.
Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.
Run the Laplace Approximation using the strong prior for both
the model 3 and model 6 formulations. Assign the results to
laplace_03_strong and
laplace_06_strong.
Confirm that the optimizations converged for both laplace approximation results.
Define the lists of required information for the strong prior.
info_03_strong <- list(
yobs = df$y,
design_matrix = X03 ,
mu_beta = 0,
tau_beta = 1,
sigma_rate = 1
)
info_06_strong <- list(
yobs = df$y,
design_matrix = X06 ,
mu_beta = 0,
tau_beta = 1,
sigma_rate = 1
)
Execute the Laplace Approximation.
laplace_03_strong <- my_laplace(rep(0, ncol(X03)+1), lm_logpost, info_03_strong)
laplace_03_strong
## $mode
## [1] 0.651611994 0.162911105 -0.152720109 -0.051739764 -0.548518942
## [6] 0.118438289 -0.079147677 0.002641276 0.017129670 -0.398729134
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0135459465 -2.873342e-04 -6.802239e-03 3.977466e-04 -5.671510e-03
## [2,] -0.0002873342 7.716870e-03 1.696316e-04 7.895995e-04 -5.465928e-04
## [3,] -0.0068022386 1.696316e-04 7.455471e-03 1.562535e-04 3.314479e-03
## [4,] 0.0003977466 7.895995e-04 1.562535e-04 8.086040e-03 5.740609e-04
## [5,] -0.0056715099 -5.465928e-04 3.314479e-03 5.740609e-04 5.582223e-03
## [6,] 0.0023593363 -3.371891e-05 1.560380e-04 5.950124e-04 -1.881045e-03
## [7,] -0.0016154431 -2.928137e-03 5.844256e-04 -1.063982e-03 1.486907e-03
## [8,] 0.0012803512 1.764089e-04 -5.707081e-04 -2.435573e-03 -1.561454e-03
## [9,] 0.0024572993 3.145148e-04 -3.430504e-03 -8.954330e-04 -2.355648e-03
## [10,] -0.0001338042 -1.539843e-05 7.467039e-05 2.359958e-06 7.772145e-05
## [,6] [,7] [,8] [,9] [,10]
## [1,] 2.359336e-03 -1.615443e-03 1.280351e-03 0.0024572993 -1.338042e-04
## [2,] -3.371891e-05 -2.928137e-03 1.764089e-04 0.0003145148 -1.539843e-05
## [3,] 1.560380e-04 5.844256e-04 -5.707081e-04 -0.0034305043 7.467039e-05
## [4,] 5.950124e-04 -1.063982e-03 -2.435573e-03 -0.0008954330 2.359958e-06
## [5,] -1.881045e-03 1.486907e-03 -1.561454e-03 -0.0023556476 7.772145e-05
## [6,] 1.197120e-02 -4.426886e-03 4.166809e-03 -0.0027336429 -4.238226e-05
## [7,] -4.426886e-03 4.913270e-03 -2.503061e-03 0.0008204165 3.291125e-05
## [8,] 4.166809e-03 -2.503061e-03 4.411329e-03 -0.0004354008 -2.628054e-05
## [9,] -2.733643e-03 8.204165e-04 -4.354008e-04 0.0034170232 -3.183560e-05
## [10,] -4.238226e-05 3.291125e-05 -2.628054e-05 -0.0000318356 5.001155e-03
##
## $log_evidence
## [1] -130.0251
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 68
laplace_06_strong <- my_laplace(rep(0, ncol(X06)+1), lm_logpost, info_06_strong)
laplace_06_strong
## $mode
## [1] 0.3211913265 -0.0287158412 -0.1003958688 0.0565332450 0.7264568887
## [6] 0.6243006390 0.1763818239 0.2158223802 0.3251240416 -0.2592733804
## [11] 0.8798073185 0.1410245934 -0.0375778966 -0.2760562036 -0.5764863280
## [16] -0.0787398641 -0.1936144200 0.4182706557 -0.1789016722 -0.1459875878
## [21] 0.3162221037 0.4518055172 0.7714941420 -0.4720480845 -0.6949870776
## [26] 0.9824389708 -0.0210462673 -0.0924572965 0.2747596768 0.4446096263
## [31] -0.1195110240 0.4518271183 0.1670136520 0.1369458446 -0.2047773966
## [36] -0.2346348028 0.1132497656 0.0004949989 -0.0504843236 0.3037346453
## [41] -0.0703279982 -0.1454908139 0.3849427236 -0.0871220868 0.0072681023
## [46] -0.3653196899 0.2061070095 0.6151077657 -0.0794043413 0.4087221759
## [51] 0.0048028222 0.0791329340 0.1492803814 0.1128845746 0.1629053375
## [56] 0.5346662507 -0.1515009892 0.2565331633 0.0343110363 0.1988994227
## [61] -0.0789994282 0.0737712445 0.1599361434 0.3559950287 -0.0149655836
## [66] -0.5995244271
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 7.388174e-02 -0.0600678247 -0.0700382386 -6.318613e-02 -0.0692512918
## [2,] -6.006782e-02 0.3030062755 -0.0522858236 9.295783e-02 0.0421114524
## [3,] -7.003824e-02 -0.0522858236 0.2342879775 -1.080196e-02 0.0966652412
## [4,] -6.318613e-02 0.0929578323 -0.0108019608 2.575234e-01 -0.0326681871
## [5,] -6.925129e-02 0.0421114524 0.0966652412 -3.266819e-02 0.2391796638
## [6,] -6.188423e-02 0.0601916798 0.0460859119 9.917829e-02 -0.0256945347
## [7,] -5.925631e-02 0.0471794638 0.0624238873 3.886402e-02 0.0824027514
## [8,] -6.645314e-02 0.0579105664 0.0632415345 6.225274e-02 0.0576271254
## [9,] -6.680026e-02 0.0549304527 0.0646731549 5.578318e-02 0.0664243770
## [10,] -6.275190e-02 0.0543955399 0.0610999064 5.587470e-02 0.0599828837
## [11,] -4.630010e-02 0.0781844267 0.0243306781 4.552251e-02 0.0412840081
## [12,] -1.202121e-01 0.0020223579 0.1647911304 8.962013e-02 0.1235546515
## [13,] -1.888208e-03 0.0606082705 -0.0249695420 1.286208e-02 -0.0015117671
## [14,] -1.550701e-03 0.0124369972 0.0009073611 -4.000300e-03 0.0041238719
## [15,] -2.782302e-02 0.0098215946 0.0149967780 1.459757e-02 0.0124782008
## [16,] 4.094782e-03 -0.0049918044 -0.0052256147 -2.691460e-05 -0.0057924597
## [17,] -1.195132e-03 0.0109292299 0.0048038525 6.512298e-03 0.0074202749
## [18,] 9.086353e-03 -0.0153095776 0.0108005984 -7.698314e-03 -0.0095012875
## [19,] -2.827570e-03 0.0005821239 0.0298062122 -2.971546e-02 0.0200128508
## [20,] -3.680405e-03 -0.0029752701 -0.0109439354 -2.754006e-02 -0.0114304446
## [21,] 9.907856e-04 -0.0172679936 0.0128424562 -2.229178e-02 0.0247057473
## [22,] 1.124633e-03 -0.0084112852 -0.0034434692 1.035710e-02 -0.0095617203
## [23,] -2.035957e-05 -0.0077620540 0.0024672814 2.489333e-03 -0.0001258530
## [24,] 2.638150e-03 -0.0102166487 -0.0035818801 6.224495e-03 -0.0108062677
## [25,] 1.424759e-03 -0.0095137904 -0.0019121183 9.613606e-04 -0.0040549482
## [26,] 2.767374e-03 -0.0089495550 -0.0002779611 1.352037e-03 -0.0015891195
## [27,] 1.155339e-02 -0.0197242645 -0.0013997207 -9.570321e-03 -0.0089019749
## [28,] -1.114971e-02 0.0229452158 -0.0043472585 2.410561e-02 0.0091316632
## [29,] 3.494102e-03 -0.0153777841 0.0094422225 -5.884072e-03 0.0002999189
## [30,] 4.593746e-03 -0.1619182966 0.0367044287 -9.230592e-03 0.0090286865
## [31,] 4.485356e-03 0.0230950510 -0.1014355663 -1.408428e-02 0.0067101644
## [32,] 2.938015e-03 0.0160786508 -0.0048567418 -1.012579e-01 0.0035767866
## [33,] 1.044547e-02 -0.0002614818 -0.0132942451 3.704924e-02 -0.1437512786
## [34,] 1.164670e-02 -0.0027002549 -0.0060281909 -2.681018e-03 0.0177471322
## [35,] 9.633549e-03 -0.0018549160 0.0016255552 -1.128571e-02 0.0106737783
## [36,] 8.234626e-03 -0.0012177722 -0.0031513396 -4.631090e-04 -0.0040585766
## [37,] 1.093352e-03 0.0009840262 0.0015898495 -4.686235e-04 0.0029220016
## [38,] 1.082618e-02 -0.0031616727 -0.0038850611 -3.792334e-03 -0.0033757556
## [39,] 1.431654e-02 -0.0065697857 -0.0125576231 -8.660753e-03 -0.0096654898
## [40,] -8.426945e-03 0.0037437271 0.0209073368 1.107747e-02 0.0153558989
## [41,] -7.156315e-03 0.0068305006 0.0026216197 6.339574e-03 0.0053641608
## [42,] 4.902059e-03 -0.0037605748 0.0069187760 -1.887681e-03 -0.0025442542
## [43,] -1.008877e-02 0.0014168834 -0.0060747527 6.190288e-03 0.0058905711
## [44,] -2.399782e-03 -0.0064290815 0.0126925609 1.846283e-02 -0.0027626719
## [45,] -7.665168e-04 0.0078584040 -0.0002285053 1.404000e-03 0.0090978271
## [46,] -5.395759e-03 0.0050030377 0.0029050765 7.664360e-03 -0.0042367623
## [47,] -2.030077e-03 0.0017628199 0.0060189298 -6.714512e-03 0.0142907578
## [48,] -9.233957e-03 0.0052731139 0.0090204842 3.281035e-03 0.0098012439
## [49,] -3.520565e-03 0.0010156825 0.0032134074 -4.241638e-04 0.0031108232
## [50,] -6.783709e-03 -0.0014608471 0.0047945220 2.922073e-03 0.0028337160
## [51,] -2.362744e-04 -0.0034572446 -0.0029227583 5.607287e-06 -0.0036541563
## [52,] -2.767269e-03 -0.0029092004 0.0084405398 -3.785341e-03 0.0055741682
## [53,] 3.821890e-03 -0.0039456232 -0.0052048283 -3.500127e-04 -0.0046631325
## [54,] 4.294682e-03 -0.0025503525 -0.0050289737 -8.162137e-03 -0.0087722711
## [55,] 6.442085e-03 -0.0060660404 0.0049222658 -1.789517e-03 -0.0181082767
## [56,] 1.337724e-02 -0.0239183300 0.0015896458 -5.216399e-03 0.0119448206
## [57,] 3.003910e-03 -0.0115227081 -0.0084600767 -8.146446e-03 0.0070699698
## [58,] 8.684649e-03 -0.0119222741 -0.0061551110 -1.740828e-02 0.0017583326
## [59,] 2.801285e-03 -0.0126097990 -0.0084595432 -2.572132e-03 -0.0231338260
## [60,] 9.763095e-03 -0.0143444079 -0.0097048353 -1.275592e-02 -0.0101778198
## [61,] 1.040441e-02 -0.0115929036 -0.0094977385 -1.184057e-02 -0.0101525183
## [62,] 5.728068e-03 -0.0093743772 -0.0065714029 -9.370131e-03 -0.0078864502
## [63,] -3.419157e-03 -0.0089431018 0.0062415693 -1.684691e-03 0.0019293485
## [64,] 2.096081e-02 -0.0085647920 -0.0323573398 -2.398484e-02 -0.0266088419
## [65,] 3.527435e-04 -0.0113280199 0.0067236546 -2.681663e-03 0.0012888016
## [66,] 1.158441e-03 -0.0001973227 -0.0015274813 -4.290852e-04 -0.0019518854
## [,6] [,7] [,8] [,9] [,10]
## [1,] -0.0618842321 -5.925631e-02 -0.0664531423 -0.0668002646 -0.0627518986
## [2,] 0.0601916798 4.717946e-02 0.0579105664 0.0549304527 0.0543955399
## [3,] 0.0460859119 6.242389e-02 0.0632415345 0.0646731549 0.0610999064
## [4,] 0.0991782890 3.886402e-02 0.0622527420 0.0557831833 0.0558746984
## [5,] -0.0256945347 8.240275e-02 0.0576271254 0.0664243770 0.0599828837
## [6,] 0.2843377225 -2.680813e-02 0.0792397798 0.0485565785 0.0599658903
## [7,] -0.0268081288 3.358085e-01 -0.0424115212 0.0935442053 0.0335976958
## [8,] 0.0792397798 -4.241152e-02 0.2434517478 -0.0090292666 0.0925853516
## [9,] 0.0485565785 9.354421e-02 -0.0090292666 0.2401879106 -0.0343087938
## [10,] 0.0599658903 3.359770e-02 0.0925853516 -0.0343087938 0.3010910887
## [11,] 0.0377474448 4.536802e-02 0.0266350269 0.0825079210 -0.0698403672
## [12,] 0.1023344255 1.045179e-01 0.1124277668 0.1162889489 0.0937935544
## [13,] 0.0054664106 -2.318981e-03 0.0107854171 -0.0178495926 0.0540534890
## [14,] 0.0053376648 -6.205929e-03 0.0108384797 0.0031806419 0.0089943057
## [15,] 0.0071112144 6.727900e-03 0.0050704655 0.0177401713 0.0024878190
## [16,] -0.0062791430 6.896358e-05 -0.0093112527 -0.0046721594 -0.0082701636
## [17,] 0.0084639520 8.482367e-03 0.0107492027 0.0044300486 0.0112036817
## [18,] -0.0111980958 -1.557830e-03 -0.0158543022 -0.0093686763 -0.0143552807
## [19,] -0.0120827662 1.158475e-02 -0.0085831051 0.0009280840 -0.0056279831
## [20,] 0.0057561899 4.738992e-03 -0.0020090119 0.0011862383 -0.0023777743
## [21,] -0.0105554312 8.035368e-03 -0.0127842187 -0.0009330341 -0.0078870767
## [22,] 0.0166622178 -9.152818e-03 -0.0005198756 -0.0046243046 -0.0062779131
## [23,] 0.0029359553 -2.477954e-02 -0.0165025412 0.0052106196 -0.0010536704
## [24,] 0.0053277025 -5.140940e-02 0.0730001585 -0.0152753226 -0.0125673969
## [25,] -0.0053437907 3.341541e-03 -0.0114209495 0.0311337997 -0.0125742214
## [26,] -0.0027149406 8.328857e-03 -0.0087220381 -0.0193559707 0.0987045231
## [27,] -0.0081743965 -9.088509e-03 -0.0032848920 -0.0162971511 -0.0088093268
## [28,] 0.0101570022 2.207109e-02 0.0052139897 0.0134734571 -0.0094601074
## [29,] -0.0021562917 2.678789e-03 -0.0047348759 0.0085627754 -0.0280918682
## [30,] 0.0029174405 6.943371e-03 0.0049699297 0.0001272717 0.0063952332
## [31,] -0.0037198462 -3.634633e-03 0.0060217631 -0.0012707472 0.0052388076
## [32,] 0.0042796508 -1.327938e-03 0.0058417926 -0.0004915936 0.0048045661
## [33,] 0.0215288163 2.370570e-03 -0.0039569092 -0.0019917259 0.0008124596
## [34,] -0.1384295313 -7.192888e-03 0.0196552425 -0.0135817665 0.0074332853
## [35,] 0.0024039577 -1.595212e-01 0.0311269218 -0.0093346513 0.0053276825
## [36,] 0.0031155289 2.193179e-02 -0.1574421779 0.0170328586 0.0001579182
## [37,] -0.0025222754 1.802122e-02 -0.0040405059 -0.1203509017 0.0137246341
## [38,] 0.0010052004 -2.454028e-03 -0.0036662901 0.0135641746 -0.1169013460
## [39,] -0.0088996486 -1.557485e-03 -0.0153605413 0.0074078858 -0.0240251969
## [40,] 0.0141422688 1.972688e-02 0.0122580207 0.0197112591 0.0012930903
## [41,] 0.0041508928 5.443640e-03 0.0049529797 0.0070004742 -0.0083791767
## [42,] 0.0023588900 -3.818215e-03 0.0048398745 -0.0023487173 0.0045659435
## [43,] 0.0107923213 3.118076e-03 0.0112549928 0.0091369241 0.0100106551
## [44,] 0.0029369298 -1.384114e-02 0.0108639596 0.0016205267 0.0057058968
## [45,] 0.0088585200 5.642291e-03 0.0061587659 0.0026979428 0.0074821252
## [46,] -0.0169018413 1.376471e-02 0.0111645001 0.0030303035 0.0073407268
## [47,] 0.0080227826 1.436263e-02 0.0016512116 0.0049870147 0.0029089920
## [48,] 0.0041290069 2.250044e-02 -0.0201317686 -0.0100194254 0.0162104633
## [49,] 0.0032943612 -1.633059e-03 -0.0186458855 0.0272998353 -0.0149281571
## [50,] 0.0024706185 1.563780e-03 -0.0016969538 0.0042296092 0.0032232805
## [51,] -0.0025860339 -5.436069e-03 -0.0014379579 -0.0028543327 -0.0231743717
## [52,] 0.0045709272 -9.244758e-04 0.0084096630 0.0038817021 -0.0031715684
## [53,] -0.0041736178 4.468101e-04 -0.0074914239 -0.0012976842 -0.0097534001
## [54,] -0.0096883269 -8.844557e-03 -0.0114954197 -0.0066478285 -0.0115796191
## [55,] -0.0075187482 -1.221456e-02 -0.0127565064 -0.0083240307 -0.0131202684
## [56,] -0.0201079600 1.903089e-03 -0.0183788508 -0.0119987824 -0.0140776200
## [57,] -0.0055884903 -2.028691e-02 -0.0077163074 -0.0073197534 -0.0113459869
## [58,] -0.0049292833 1.853078e-02 -0.0221427761 -0.0072825735 -0.0147321550
## [59,] 0.0118005917 -7.632556e-03 -0.0057542892 -0.0089844422 -0.0089625452
## [60,] -0.0148935874 -8.274553e-03 0.0203523546 0.0051728290 -0.0162957306
## [61,] -0.0110782960 -1.280079e-02 0.0151152760 -0.0149840042 -0.0028193023
## [62,] -0.0078359324 -9.636796e-03 -0.0086588268 -0.0013665130 -0.0074097083
## [63,] 0.0000881407 -3.332741e-04 0.0021410759 -0.0049519681 0.0182957365
## [64,] -0.0233372402 -2.830553e-02 -0.0263821304 -0.0254267385 -0.0111549692
## [65,] 0.0024200813 -3.141391e-03 0.0025387744 -0.0009411105 0.0163543524
## [66,] -0.0017369352 -1.938593e-03 -0.0005927517 -0.0016449567 -0.0003278450
## [,11] [,12] [,13] [,14] [,15]
## [1,] -4.630010e-02 -0.120212105 -1.888208e-03 -0.0015507012 -0.0278230167
## [2,] 7.818443e-02 0.002022358 6.060827e-02 0.0124369972 0.0098215946
## [3,] 2.433068e-02 0.164791130 -2.496954e-02 0.0009073611 0.0149967780
## [4,] 4.552251e-02 0.089620132 1.286208e-02 -0.0040003001 0.0145975651
## [5,] 4.128401e-02 0.123554652 -1.511767e-03 0.0041238719 0.0124782008
## [6,] 3.774744e-02 0.102334426 5.466411e-03 0.0053376648 0.0071112144
## [7,] 4.536802e-02 0.104517899 -2.318981e-03 -0.0062059292 0.0067279003
## [8,] 2.663503e-02 0.112427767 1.078542e-02 0.0108384797 0.0050704655
## [9,] 8.250792e-02 0.116288949 -1.784959e-02 0.0031806419 0.0177401713
## [10,] -6.984037e-02 0.093793554 5.405349e-02 0.0089943057 0.0024878190
## [11,] 2.416619e-01 0.013250525 -8.640185e-02 0.0043422160 0.0219562579
## [12,] 1.325052e-02 0.450198107 5.776840e-02 -0.0067400832 0.0046553810
## [13,] -8.640185e-02 0.057768397 3.824925e-01 -0.0002533215 -0.0074295101
## [14,] 4.342216e-03 -0.006740083 -2.533215e-04 0.0859640886 -0.0082749776
## [15,] 2.195626e-02 0.004655381 -7.429510e-03 -0.0082749776 0.1267406308
## [16,] -2.242863e-03 -0.006367182 -1.990767e-03 -0.0293026218 0.0113353104
## [17,] -8.718037e-04 0.020088214 3.200534e-03 0.0113530073 -0.0486897302
## [18,] -1.389449e-02 0.008574372 -9.761102e-03 -0.0723826903 0.0014942587
## [19,] 3.291219e-03 -0.003098193 5.676872e-03 -0.0637733794 0.0209356928
## [20,] -1.132256e-03 0.012779736 -1.206851e-03 -0.0558131593 0.0116898527
## [21,] -3.893305e-03 0.008095030 -1.226140e-03 -0.0733337969 0.0077738231
## [22,] -2.826239e-03 0.004869188 4.895345e-05 -0.0646529070 0.0049323286
## [23,] -4.030865e-03 0.012084257 1.944733e-03 -0.0596173387 -0.0079445861
## [24,] -2.544774e-03 0.003513290 -1.233151e-03 -0.0688708586 0.0051272267
## [25,] -5.847191e-03 0.003100646 6.419829e-03 -0.0558245913 0.0157939853
## [26,] -3.400507e-02 -0.007986949 -2.962640e-03 -0.0556023669 -0.0122767656
## [27,] -3.098056e-02 0.018985631 3.613324e-02 -0.0184609335 -0.0195814608
## [28,] 3.212559e-02 -0.003024520 -2.186449e-02 -0.1007318925 -0.0206001452
## [29,] 3.744990e-02 -0.025204091 -1.108691e-01 0.0030965408 -0.0155605922
## [30,] -5.394854e-03 0.015553422 1.294222e-03 0.0055631356 -0.0608853627
## [31,] -2.994614e-03 0.002717528 6.280245e-03 0.0128117540 -0.0477657954
## [32,] 1.047671e-03 -0.001101974 7.304802e-03 0.0074103651 -0.0402092121
## [33,] -8.593447e-03 0.006152289 3.132242e-03 0.0072944847 -0.0678389568
## [34,] -1.328031e-02 0.008578176 5.115637e-03 0.0023922016 -0.0788817554
## [35,] -9.418109e-03 0.011657290 5.331985e-03 -0.0006088929 -0.0774128550
## [36,] -7.860600e-03 0.005379019 4.408843e-03 -0.0036521661 -0.0548534911
## [37,] 8.009089e-05 0.005644747 6.421357e-04 0.0035081909 -0.0228267628
## [38,] -7.213859e-03 0.008584889 9.105655e-03 -0.0019546805 -0.0672228464
## [39,] -1.172716e-01 0.027128033 2.998261e-02 -0.0017470598 -0.0379103653
## [40,] 3.602594e-02 -0.112521356 -5.914511e-03 -0.0247461147 -0.0615969155
## [41,] 2.566126e-02 -0.004501804 -1.973709e-01 -0.0036783861 0.0131163041
## [42,] -1.381262e-02 0.022512507 -8.496759e-03 0.0229422006 -0.0377263264
## [43,] 8.458107e-03 0.006553870 3.295950e-03 0.0246423117 0.0073630150
## [44,] -1.840346e-03 0.010767684 -1.168503e-03 0.0148525847 -0.0103746417
## [45,] 9.140833e-04 0.002129326 4.586386e-03 0.0278850284 -0.0240583016
## [46,] 2.016064e-03 0.008019441 1.777548e-03 0.0146160333 -0.0027477220
## [47,] 2.256809e-03 0.002129469 2.138501e-03 0.0180895855 -0.0072691282
## [48,] 4.275278e-03 0.008960051 -6.985036e-04 0.0136578509 0.0073137285
## [49,] 1.045169e-02 0.006850435 4.017028e-03 0.0098286963 0.0078062103
## [50,] 6.626123e-03 -0.003787212 -1.423081e-02 -0.0139567170 0.0268346569
## [51,] 1.789227e-03 -0.001091437 1.484803e-02 -0.0144450887 0.0205642605
## [52,] 6.548378e-03 0.008358445 -2.077289e-02 0.0137339646 -0.0112653098
## [53,] 1.664028e-02 -0.027009221 -4.007051e-02 -0.0203988331 -0.0024845039
## [54,] -6.009452e-03 -0.010353641 -7.555466e-03 -0.0146403860 0.0357974833
## [55,] -1.745415e-03 -0.026325630 -9.902701e-04 -0.0102225216 0.0324782154
## [56,] -9.588607e-03 -0.021963620 -2.010303e-03 -0.0114688173 0.0018715350
## [57,] -9.126114e-04 -0.021347937 -2.891586e-03 -0.0116540697 0.0440417747
## [58,] -2.966724e-03 -0.026424031 -2.421527e-04 -0.0136186344 0.0221776199
## [59,] -1.437684e-03 -0.019370601 -3.812035e-03 -0.0007215787 0.0415662736
## [60,] -7.346019e-03 -0.025968068 -3.332772e-03 -0.0059461596 0.0155788496
## [61,] -1.694378e-03 -0.014597599 6.412798e-03 0.0043901875 0.0003724539
## [62,] 5.255333e-03 -0.016302550 -1.735602e-03 0.0072478160 0.0176508293
## [63,] 1.595621e-02 -0.011284697 -1.017475e-02 0.0052923990 0.0190135027
## [64,] -2.098268e-02 -0.030288776 9.938115e-03 0.0045922966 0.0347588717
## [65,] -1.222436e-02 0.013875413 5.171887e-02 0.0104690932 -0.0104445371
## [66,] -2.822354e-03 -0.001529455 1.338728e-03 0.0009941847 0.0006513935
## [,16] [,17] [,18] [,19] [,20]
## [1,] 4.094782e-03 -0.0011951320 0.009086353 -0.0028275701 -3.680405e-03
## [2,] -4.991804e-03 0.0109292299 -0.015309578 0.0005821239 -2.975270e-03
## [3,] -5.225615e-03 0.0048038525 0.010800598 0.0298062122 -1.094394e-02
## [4,] -2.691460e-05 0.0065122984 -0.007698314 -0.0297154609 -2.754006e-02
## [5,] -5.792460e-03 0.0074202749 -0.009501287 0.0200128508 -1.143044e-02
## [6,] -6.279143e-03 0.0084639520 -0.011198096 -0.0120827662 5.756190e-03
## [7,] 6.896358e-05 0.0084823673 -0.001557830 0.0115847521 4.738992e-03
## [8,] -9.311253e-03 0.0107492027 -0.015854302 -0.0085831051 -2.009012e-03
## [9,] -4.672159e-03 0.0044300486 -0.009368676 0.0009280840 1.186238e-03
## [10,] -8.270164e-03 0.0112036817 -0.014355281 -0.0056279831 -2.377774e-03
## [11,] -2.242863e-03 -0.0008718037 -0.013894494 0.0032912195 -1.132256e-03
## [12,] -6.367182e-03 0.0200882138 0.008574372 -0.0030981933 1.277974e-02
## [13,] -1.990767e-03 0.0032005338 -0.009761102 0.0056768725 -1.206851e-03
## [14,] -2.930262e-02 0.0113530073 -0.072382690 -0.0637733794 -5.581316e-02
## [15,] 1.133531e-02 -0.0486897302 0.001494259 0.0209356928 1.168985e-02
## [16,] 5.891221e-02 0.0170854219 0.023482779 0.0157191923 1.498618e-02
## [17,] 1.708542e-02 0.0512068666 -0.014436508 -0.0178858429 -1.195679e-02
## [18,] 2.348278e-02 -0.0144365076 0.480181004 -0.0585125473 6.542216e-02
## [19,] 1.571919e-02 -0.0178858429 -0.058512547 0.4844080291 -9.009648e-02
## [20,] 1.498618e-02 -0.0119567917 0.065422163 -0.0900964799 5.626614e-01
## [21,] 2.156565e-02 -0.0121956775 0.060240730 0.0824860422 -6.463277e-02
## [22,] 1.584611e-02 -0.0112438228 0.056023191 0.0402282436 6.954319e-02
## [23,] 1.465323e-02 -0.0038869686 0.052641499 0.0412472492 3.334743e-02
## [24,] 1.954385e-02 -0.0100303757 0.057213437 0.0496040384 4.939204e-02
## [25,] 1.495279e-02 -0.0142744162 0.045340962 0.0447529726 3.631935e-02
## [26,] 4.908235e-03 -0.0051026564 0.049609404 0.0395007927 3.608538e-02
## [27,] 3.704836e-03 0.0068874954 0.072043312 -0.0024751030 1.299357e-02
## [28,] 9.392049e-03 -0.0104149603 -0.053708209 0.1099006161 6.136106e-02
## [29,] 3.358936e-03 0.0099570799 0.079613244 -0.0259508334 -4.125468e-04
## [30,] -1.917023e-02 0.0157670309 -0.038591427 0.0122647916 -2.865601e-02
## [31,] -1.524091e-02 0.0131051615 -0.011510792 0.1078442634 1.628665e-04
## [32,] -1.465103e-02 0.0097744916 -0.021147647 0.0138836179 4.617744e-02
## [33,] -1.295221e-02 0.0215030862 0.001736177 -0.0304483019 -1.132455e-02
## [34,] -6.601048e-03 0.0288207595 0.005230445 -0.0220583287 -6.516430e-03
## [35,] -6.223703e-03 0.0289386748 0.002113749 -0.0035056143 -4.358827e-03
## [36,] -1.079727e-03 0.0187974085 0.004487630 -0.0025084057 2.158262e-03
## [37,] -4.485426e-03 0.0049007326 -0.000862003 -0.0050592640 -3.170161e-03
## [38,] -6.637053e-03 0.0215371637 0.005572699 -0.0061992115 -1.484772e-03
## [39,] 4.027293e-03 0.0180536258 0.004284863 0.0004062804 -1.872899e-03
## [40,] -2.458978e-02 -0.0035174353 0.022331037 0.0050197315 1.762944e-02
## [41,] 2.394066e-03 0.0011179853 0.002857199 0.0101627476 2.407180e-03
## [42,] -3.928317e-02 0.0042849727 -0.188175599 -0.0129189541 -7.419199e-03
## [43,] -6.105475e-02 -0.0271050894 0.017824991 -0.1057689068 3.504479e-02
## [44,] -4.937853e-02 -0.0172295521 -0.006616627 0.0061485308 -1.990570e-01
## [45,] -5.836631e-02 -0.0110432953 -0.026451958 -0.0094477197 2.684578e-02
## [46,] -4.653007e-02 -0.0176756024 -0.010844617 -0.0119473014 4.605268e-03
## [47,] -5.358822e-02 -0.0200813334 -0.017514535 0.0055305877 -2.195500e-02
## [48,] -5.162043e-02 -0.0258769180 -0.010034121 -0.0024342443 -3.774976e-03
## [49,] -3.559688e-02 -0.0211593528 -0.007697566 -0.0004587154 -3.969198e-03
## [50,] -2.825887e-02 -0.0340256375 0.013270253 0.0182068438 1.385489e-02
## [51,] -6.045598e-04 -0.0258180468 -0.010795949 0.0171428330 1.173162e-02
## [52,] -8.375842e-02 -0.0218082657 0.048488043 -0.0059466407 -3.704134e-03
## [53,] 2.049146e-02 -0.0104305405 -0.013801658 0.0161600877 1.268248e-02
## [54,] -1.020895e-02 -0.0374298216 -0.021110350 0.0075661163 1.966409e-02
## [55,] -1.795295e-02 -0.0457777897 0.025338258 -0.0089003024 1.686015e-02
## [56,] -1.491381e-02 -0.0287687695 0.016577960 0.0220284103 -3.156180e-02
## [57,] -1.680979e-02 -0.0495189257 0.016294360 0.0130501036 2.071868e-02
## [58,] -1.390964e-02 -0.0372088617 0.009952941 0.0341604678 1.951720e-02
## [59,] -2.495776e-02 -0.0501925430 0.011715849 -0.0049922760 1.361234e-03
## [60,] -2.232410e-02 -0.0379917766 0.011260732 0.0106828093 6.223554e-03
## [61,] -2.694200e-02 -0.0275585569 0.001557646 0.0006178148 -5.271324e-04
## [62,] -3.115579e-02 -0.0376624019 -0.001210974 0.0017050247 -1.098122e-03
## [63,] -2.384871e-02 -0.0224721667 -0.008640609 -0.0023998052 -4.518773e-06
## [64,] -5.142726e-02 -0.0852791175 0.024885820 0.0185187425 2.249966e-03
## [65,] -3.254530e-02 -0.0057663702 -0.016942081 -0.0104712607 -3.865148e-03
## [66,] 1.170427e-03 0.0007823946 -0.003350941 0.0004903678 -1.845255e-04
## [,21] [,22] [,23] [,24] [,25]
## [1,] 0.0009907856 1.124633e-03 -2.035957e-05 0.0026381503 0.0014247592
## [2,] -0.0172679936 -8.411285e-03 -7.762054e-03 -0.0102166487 -0.0095137904
## [3,] 0.0128424562 -3.443469e-03 2.467281e-03 -0.0035818801 -0.0019121183
## [4,] -0.0222917817 1.035710e-02 2.489333e-03 0.0062244946 0.0009613606
## [5,] 0.0247057473 -9.561720e-03 -1.258530e-04 -0.0108062677 -0.0040549482
## [6,] -0.0105554312 1.666222e-02 2.935955e-03 0.0053277025 -0.0053437907
## [7,] 0.0080353681 -9.152818e-03 -2.477954e-02 -0.0514093951 0.0033415414
## [8,] -0.0127842187 -5.198756e-04 -1.650254e-02 0.0730001585 -0.0114209495
## [9,] -0.0009330341 -4.624305e-03 5.210620e-03 -0.0152753226 0.0311337997
## [10,] -0.0078870767 -6.277913e-03 -1.053670e-03 -0.0125673969 -0.0125742214
## [11,] -0.0038933048 -2.826239e-03 -4.030865e-03 -0.0025447742 -0.0058471907
## [12,] 0.0080950303 4.869188e-03 1.208426e-02 0.0035132898 0.0031006464
## [13,] -0.0012261404 4.895345e-05 1.944733e-03 -0.0012331507 0.0064198293
## [14,] -0.0733337969 -6.465291e-02 -5.961734e-02 -0.0688708586 -0.0558245913
## [15,] 0.0077738231 4.932329e-03 -7.944586e-03 0.0051272267 0.0157939853
## [16,] 0.0215656546 1.584611e-02 1.465323e-02 0.0195438537 0.0149527911
## [17,] -0.0121956775 -1.124382e-02 -3.886969e-03 -0.0100303757 -0.0142744162
## [18,] 0.0602407299 5.602319e-02 5.264150e-02 0.0572134374 0.0453409624
## [19,] 0.0824860422 4.022824e-02 4.124725e-02 0.0496040384 0.0447529726
## [20,] -0.0646327738 6.954319e-02 3.334743e-02 0.0493920363 0.0363193454
## [21,] 0.3985571004 -7.230001e-02 8.053273e-02 0.0439778406 0.0515360732
## [22,] -0.0723000074 4.814415e-01 -8.584290e-02 0.0732966952 0.0361673503
## [23,] 0.0805327263 -8.584290e-02 4.941713e-01 -0.0582452134 0.0689991915
## [24,] 0.0439778406 7.329670e-02 -5.824521e-02 0.4048233331 -0.0618114577
## [25,] 0.0515360732 3.616735e-02 6.899919e-02 -0.0618114577 0.4876343961
## [26,] 0.0479595615 4.708846e-02 3.269253e-02 0.0597297155 -0.0732910320
## [27,] 0.0159769239 1.250437e-02 1.988358e-02 0.0091630264 0.0526227697
## [28,] 0.0871876561 7.778600e-02 7.689093e-02 0.0809297547 0.0796577844
## [29,] -0.0024463305 -2.387865e-03 -4.613509e-04 -0.0024938603 -0.0093604826
## [30,] 0.0016888559 -3.303895e-03 4.028646e-03 -0.0052880002 -0.0094125152
## [31,] -0.0101830899 -8.449330e-03 2.852326e-03 -0.0080119326 -0.0108520621
## [32,] -0.0222853652 -7.446415e-03 3.933325e-03 -0.0047237127 -0.0088407907
## [33,] 0.0103195822 1.435094e-02 1.362744e-02 -0.0069730809 -0.0106290526
## [34,] -0.0009931488 1.453565e-02 -1.947479e-02 0.0130951406 -0.0084623828
## [35,] 0.0042627497 -2.543908e-02 -2.686771e-02 -0.0039974653 -0.0052469471
## [36,] -0.0037025730 1.963284e-03 -2.724693e-02 0.0068827341 0.0086981338
## [37,] -0.0029485427 -5.134531e-04 -1.138932e-02 -0.0049068814 0.0757105992
## [38,] 0.0012262941 3.225658e-03 1.244347e-02 0.0017671286 -0.0029278292
## [39,] 0.0003825325 9.223576e-04 1.405693e-03 0.0045085646 -0.0056235981
## [40,] 0.0235025147 2.293774e-02 2.740832e-02 0.0217173079 0.0129604221
## [41,] 0.0031804235 1.904098e-03 2.249320e-03 0.0023583189 0.0082140283
## [42,] -0.0203085896 -1.237928e-02 -9.964897e-03 -0.0139907556 -0.0136775373
## [43,] -0.0224165809 -1.151140e-02 -1.288155e-02 -0.0167458254 -0.0103490331
## [44,] 0.0144632492 -6.016985e-03 6.602718e-03 -0.0032192523 -0.0069999094
## [45,] -0.1161064054 2.394903e-02 -2.986026e-02 -0.0198362859 -0.0146233312
## [46,] -0.0246728176 -2.075274e-01 5.568316e-02 0.0165328522 -0.0125805671
## [47,] 0.0299041400 4.182429e-02 -2.160146e-01 0.0041047761 -0.0096219537
## [48,] -0.0089357141 -8.080836e-03 2.013336e-03 -0.1933540471 0.0349096728
## [49,] -0.0014176109 -1.289925e-02 3.877026e-02 0.0272876984 -0.1724487185
## [50,] 0.0136584744 1.669819e-02 4.506044e-03 0.0170933206 0.0073659712
## [51,] 0.0128682642 1.056324e-02 1.133392e-02 0.0072910347 0.0412338129
## [52,] -0.0054391931 -9.103331e-04 1.046768e-03 -0.0059436468 0.0071491864
## [53,] 0.0164749288 1.398663e-02 1.228453e-02 0.0153821656 0.0066937745
## [54,] 0.0129775794 1.331842e-02 6.685237e-03 0.0131984257 0.0152347157
## [55,] 0.0091310321 1.123347e-02 5.313420e-03 0.0091113030 0.0120282075
## [56,] 0.0219066903 8.841205e-03 -2.892289e-03 0.0090327713 0.0104892576
## [57,] -0.0021542478 1.372011e-02 6.851414e-03 0.0111463202 0.0145355131
## [58,] -0.0065757252 -4.321209e-02 1.207838e-02 0.0191879069 0.0095008979
## [59,] 0.0093395539 6.564501e-02 3.978848e-02 -0.0006985245 0.0107797611
## [60,] 0.0095024519 2.585545e-03 3.146281e-02 0.0372460444 -0.0125690922
## [61,] -0.0021158524 1.479359e-03 -1.982642e-02 -0.0089147907 0.0305692711
## [62,] -0.0033458448 -2.520089e-03 -3.438155e-03 -0.0057297335 0.0120085659
## [63,] -0.0021543553 -1.026163e-03 -4.365230e-03 -0.0016012053 -0.0139470074
## [64,] 0.0007927347 3.639094e-03 -5.865609e-03 0.0006104786 -0.0033836824
## [65,] -0.0076411454 -3.111429e-03 -5.038739e-03 -0.0043066024 -0.0116242270
## [66,] -0.0020663603 -2.290041e-03 -3.596957e-03 0.0017221013 0.0024711107
## [,26] [,27] [,28] [,29] [,30]
## [1,] 0.0027673742 0.0115533910 -0.0111497072 0.0034941016 0.0045937458
## [2,] -0.0089495550 -0.0197242645 0.0229452158 -0.0153777841 -0.1619182966
## [3,] -0.0002779611 -0.0013997207 -0.0043472585 0.0094422225 0.0367044287
## [4,] 0.0013520371 -0.0095703213 0.0241056135 -0.0058840722 -0.0092305918
## [5,] -0.0015891195 -0.0089019749 0.0091316632 0.0002999189 0.0090286865
## [6,] -0.0027149406 -0.0081743965 0.0101570022 -0.0021562917 0.0029174405
## [7,] 0.0083288566 -0.0090885088 0.0220710929 0.0026787888 0.0069433707
## [8,] -0.0087220381 -0.0032848920 0.0052139897 -0.0047348759 0.0049699297
## [9,] -0.0193559707 -0.0162971511 0.0134734571 0.0085627754 0.0001272717
## [10,] 0.0987045231 -0.0088093268 -0.0094601074 -0.0280918682 0.0063952332
## [11,] -0.0340050710 -0.0309805640 0.0321255918 0.0374498972 -0.0053948540
## [12,] -0.0079869495 0.0189856310 -0.0030245197 -0.0252040906 0.0155534218
## [13,] -0.0029626398 0.0361332391 -0.0218644873 -0.1108691388 0.0012942225
## [14,] -0.0556023669 -0.0184609335 -0.1007318925 0.0030965408 0.0055631356
## [15,] -0.0122767656 -0.0195814608 -0.0206001452 -0.0155605922 -0.0608853627
## [16,] 0.0049082348 0.0037048364 0.0093920492 0.0033589359 -0.0191702274
## [17,] -0.0051026564 0.0068874954 -0.0104149603 0.0099570799 0.0157670309
## [18,] 0.0496094040 0.0720433116 -0.0537082088 0.0796132437 -0.0385914268
## [19,] 0.0395007927 -0.0024751030 0.1099006161 -0.0259508334 0.0122647916
## [20,] 0.0360853775 0.0129935658 0.0613610584 -0.0004125468 -0.0286560089
## [21,] 0.0479595615 0.0159769239 0.0871876561 -0.0024463305 0.0016888559
## [22,] 0.0470884645 0.0125043665 0.0777860046 -0.0023878649 -0.0033038950
## [23,] 0.0326925274 0.0198835776 0.0768909281 -0.0004613509 0.0040286455
## [24,] 0.0597297155 0.0091630264 0.0809297547 -0.0024938603 -0.0052880002
## [25,] -0.0732910320 0.0526227697 0.0796577844 -0.0093604826 -0.0094125152
## [26,] 0.5980027229 -0.1487519180 0.0124021025 0.0102313954 0.0112841069
## [27,] -0.1487519180 0.5382017204 0.0350982133 -0.0517673562 -0.0044704012
## [28,] 0.0124021025 0.0350982133 0.6082581194 0.0505175496 0.0420197973
## [29,] 0.0102313954 -0.0517673562 0.0505175496 0.5641650697 -0.0085175352
## [30,] 0.0112841069 -0.0044704012 0.0420197973 -0.0085175352 0.6906921937
## [31,] -0.0017489915 0.0045198316 0.0031584389 0.0036600539 -0.0414641356
## [32,] 0.0051159394 0.0008411301 0.0122665800 0.0009755946 0.0355438112
## [33,] 0.0060371345 0.0096917854 0.0103025049 0.0088827417 0.0209056209
## [34,] 0.0082583224 0.0130351037 0.0160752077 0.0089668076 0.0311246254
## [35,] 0.0103987444 0.0139348242 0.0235671169 0.0097591476 0.0396415388
## [36,] 0.0022365789 0.0117907055 0.0195347584 0.0049537801 0.0270034586
## [37,] -0.0002276930 0.0014974287 0.0062570799 0.0082399602 0.0124481667
## [38,] 0.0449844652 -0.0358105154 -0.0033909017 -0.0004049058 0.0345797344
## [39,] -0.0388382714 -0.0911689952 0.0075890933 0.0479822649 0.0453725715
## [40,] 0.0074691652 0.0033283251 -0.0109485467 -0.0517418813 -0.0319464189
## [41,] -0.0116178887 0.0281650620 -0.0588829307 -0.1644545433 0.0360487396
## [42,] -0.0040663301 -0.0145383873 0.0391820595 -0.0243602304 -0.1611306876
## [43,] -0.0023064948 -0.0025472251 -0.0149110721 -0.0006268913 0.0723154726
## [44,] 0.0028604159 0.0008563228 0.0025090579 -0.0017684343 0.0060363454
## [45,] -0.0023132216 -0.0020198597 -0.0031365303 -0.0030199864 0.0328502067
## [46,] 0.0022899921 -0.0021512780 0.0025695888 -0.0046690845 0.0094760142
## [47,] 0.0002724940 -0.0010534224 0.0041811948 -0.0038088680 0.0228732917
## [48,] 0.0245682123 -0.0160033611 -0.0010642582 -0.0061556046 0.0116456946
## [49,] -0.0331357086 0.0278007265 0.0208301705 0.0061747921 0.0054867057
## [50,] -0.2028869534 -0.0780762790 0.0041676316 0.0092415104 -0.0026148193
## [51,] -0.0639558675 -0.1954138897 0.0284947390 0.0287180974 -0.0101130528
## [52,] -0.0070587637 -0.0030374178 -0.1437866164 0.0183109324 0.0297966455
## [53,] 0.0195491747 0.0028111971 0.0298810586 -0.1587036296 -0.0031878315
## [54,] 0.0069614386 -0.0008741449 0.0104048649 -0.0053938553 -0.1639789470
## [55,] 0.0067163111 -0.0044091636 0.0118247514 -0.0075883607 -0.0030360204
## [56,] 0.0103639422 0.0018058184 0.0180973595 -0.0032965339 0.0136201360
## [57,] 0.0054380730 -0.0055586039 0.0111261284 -0.0092681686 -0.0169344433
## [58,] 0.0107580603 -0.0035121117 0.0196120035 -0.0077917067 0.0102696303
## [59,] -0.0014299856 -0.0055745424 -0.0003620087 -0.0073863110 -0.0225637508
## [60,] 0.0111751585 -0.0118017857 0.0036731127 -0.0130438428 0.0007975820
## [61,] -0.0130678251 0.0293778669 0.0284116775 0.0161002331 0.0063786523
## [62,] 0.0318297449 0.0575277358 0.0157409355 -0.0217203074 0.0001250982
## [63,] 0.0582839880 0.0515255017 0.0037412315 -0.0203014894 -0.0162726597
## [64,] 0.0384247868 -0.0030166177 0.0263098612 -0.0169109756 0.0355385629
## [65,] 0.0251602559 -0.0111205966 -0.0001738525 0.0243445610 -0.0082050842
## [66,] -0.0062966488 0.0022280846 -0.0006468791 -0.0020580772 -0.0041365886
## [,31] [,32] [,33] [,34] [,35]
## [1,] 0.0044853558 0.0029380153 1.044547e-02 0.0116466967 0.0096335492
## [2,] 0.0230950510 0.0160786508 -2.614818e-04 -0.0027002549 -0.0018549160
## [3,] -0.1014355663 -0.0048567418 -1.329425e-02 -0.0060281909 0.0016255552
## [4,] -0.0140842772 -0.1012578599 3.704924e-02 -0.0026810183 -0.0112857102
## [5,] 0.0067101644 0.0035767866 -1.437513e-01 0.0177471322 0.0106737783
## [6,] -0.0037198462 0.0042796508 2.152882e-02 -0.1384295313 0.0024039577
## [7,] -0.0036346329 -0.0013279379 2.370570e-03 -0.0071928879 -0.1595211925
## [8,] 0.0060217631 0.0058417926 -3.956909e-03 0.0196552425 0.0311269218
## [9,] -0.0012707472 -0.0004915936 -1.991726e-03 -0.0135817665 -0.0093346513
## [10,] 0.0052388076 0.0048045661 8.124596e-04 0.0074332853 0.0053276825
## [11,] -0.0029946137 0.0010476705 -8.593447e-03 -0.0132803122 -0.0094181090
## [12,] 0.0027175277 -0.0011019742 6.152289e-03 0.0085781759 0.0116572902
## [13,] 0.0062802453 0.0073048015 3.132242e-03 0.0051156366 0.0053319852
## [14,] 0.0128117540 0.0074103651 7.294485e-03 0.0023922016 -0.0006088929
## [15,] -0.0477657954 -0.0402092121 -6.783896e-02 -0.0788817554 -0.0774128550
## [16,] -0.0152409130 -0.0146510256 -1.295221e-02 -0.0066010485 -0.0062237028
## [17,] 0.0131051615 0.0097744916 2.150309e-02 0.0288207595 0.0289386748
## [18,] -0.0115107922 -0.0211476471 1.736177e-03 0.0052304452 0.0021137489
## [19,] 0.1078442634 0.0138836179 -3.044830e-02 -0.0220583287 -0.0035056143
## [20,] 0.0001628665 0.0461774411 -1.132455e-02 -0.0065164297 -0.0043588265
## [21,] -0.0101830899 -0.0222853652 1.031958e-02 -0.0009931488 0.0042627497
## [22,] -0.0084493296 -0.0074464149 1.435094e-02 0.0145356496 -0.0254390777
## [23,] 0.0028523257 0.0039333248 1.362744e-02 -0.0194747871 -0.0268677147
## [24,] -0.0080119326 -0.0047237127 -6.973081e-03 0.0130951406 -0.0039974653
## [25,] -0.0108520621 -0.0088407907 -1.062905e-02 -0.0084623828 -0.0052469471
## [26,] -0.0017489915 0.0051159394 6.037134e-03 0.0082583224 0.0103987444
## [27,] 0.0045198316 0.0008411301 9.691785e-03 0.0130351037 0.0139348242
## [28,] 0.0031584389 0.0122665800 1.030250e-02 0.0160752077 0.0235671169
## [29,] 0.0036600539 0.0009755946 8.882742e-03 0.0089668076 0.0097591476
## [30,] -0.0414641356 0.0355438112 2.090562e-02 0.0311246254 0.0396415388
## [31,] 0.7066038077 -0.0816296942 6.195210e-02 0.0495237818 0.0268668300
## [32,] -0.0816296942 0.7411227112 -2.839544e-02 0.0479617490 0.0190534923
## [33,] 0.0619521025 -0.0283954407 5.884981e-01 -0.1015384458 0.0761694899
## [34,] 0.0495237818 0.0479617490 -1.015384e-01 0.6023001486 -0.0929277852
## [35,] 0.0268668300 0.0190534923 7.616949e-02 -0.0929277852 0.5725737930
## [36,] 0.0167813196 0.0162055485 2.034672e-02 0.0402636448 -0.0596553258
## [37,] 0.0109084789 0.0085915660 1.651588e-02 0.0105089959 0.0248370459
## [38,] 0.0251236150 0.0225762354 3.515861e-02 0.0454029999 0.0348759245
## [39,] 0.0139753349 0.0089584699 2.017776e-02 0.0223047686 0.0230232331
## [40,] 0.0252053578 0.0285060995 3.782349e-02 0.0415821516 0.0444258301
## [41,] -0.0037278812 -0.0058303273 -6.471458e-03 -0.0087249303 -0.0060194816
## [42,] -0.0333533923 0.0175478879 2.787713e-02 0.0255080630 0.0219887804
## [43,] 0.0574846357 0.0387729789 3.999757e-03 -0.0088778947 -0.0044334134
## [44,] 0.0397264926 -0.1000291693 6.025124e-02 0.0582120943 -0.0045916047
## [45,] -0.0002094054 0.0371089727 -5.414571e-02 -0.0091270431 0.0164777512
## [46,] 0.0134744514 0.0146169145 1.517315e-02 -0.0044497593 0.0221301988
## [47,] -0.0027339545 0.0059869348 -3.066635e-02 0.0433687569 0.0337406398
## [48,] 0.0054146579 0.0087012249 2.841949e-03 -0.0119020243 -0.0037179782
## [49,] 0.0032136408 0.0040152188 -1.301777e-03 -0.0042483553 -0.0150233843
## [50,] -0.0057108023 -0.0015654409 -1.067431e-02 -0.0140427890 -0.0188420296
## [51,] -0.0108466575 -0.0054220985 -9.145367e-03 -0.0099459176 -0.0113952865
## [52,] 0.0285635859 0.0179419961 1.355682e-02 0.0052046923 0.0066780226
## [53,] -0.0082864502 -0.0008506838 1.810981e-03 0.0036596443 0.0041946220
## [54,] -0.0013752159 -0.0151563802 -1.561357e-02 -0.0204977955 -0.0211957510
## [55,] -0.1724986504 0.0391261259 -7.191055e-03 -0.0177997091 -0.0202140588
## [56,] -0.0040413081 -0.2427381226 -5.343302e-02 -0.0369794743 0.0096030124
## [57,] -0.0019875815 0.0204712934 -1.302130e-01 0.0344121973 -0.0426987724
## [58,] -0.0451500487 -0.0180597694 -3.157139e-02 -0.1937800008 0.0435562481
## [59,] 0.0227605217 0.0027094843 3.468068e-02 0.0158360680 -0.2003814410
## [60,] -0.0010816742 0.0007174240 -4.129133e-03 -0.0092968329 0.0162173107
## [61,] 0.0074810063 0.0057913246 7.480836e-03 -0.0041563878 0.0176569382
## [62,] 0.0012658839 0.0014271605 -3.785482e-03 -0.0105087253 -0.0102770638
## [63,] -0.0067505679 -0.0006721685 -6.847785e-03 -0.0129389634 -0.0094132852
## [64,] 0.0114439363 0.0035841644 -6.748054e-03 -0.0181538508 -0.0229148853
## [65,] 0.0023271543 0.0084661544 6.995568e-03 0.0074793219 0.0018986146
## [66,] 0.0012970194 -0.0026372534 -8.377401e-05 0.0002258945 0.0007066990
## [,36] [,37] [,38] [,39] [,40]
## [1,] 0.0082346259 1.093352e-03 0.0108261821 1.431654e-02 -0.008426945
## [2,] -0.0012177722 9.840262e-04 -0.0031616727 -6.569786e-03 0.003743727
## [3,] -0.0031513396 1.589849e-03 -0.0038850611 -1.255762e-02 0.020907337
## [4,] -0.0004631090 -4.686235e-04 -0.0037923339 -8.660753e-03 0.011077474
## [5,] -0.0040585766 2.922002e-03 -0.0033757556 -9.665490e-03 0.015355899
## [6,] 0.0031155289 -2.522275e-03 0.0010052004 -8.899649e-03 0.014142269
## [7,] 0.0219317933 1.802122e-02 -0.0024540278 -1.557485e-03 0.019726879
## [8,] -0.1574421779 -4.040506e-03 -0.0036662901 -1.536054e-02 0.012258021
## [9,] 0.0170328586 -1.203509e-01 0.0135641746 7.407886e-03 0.019711259
## [10,] 0.0001579182 1.372463e-02 -0.1169013460 -2.402520e-02 0.001293090
## [11,] -0.0078606003 8.009089e-05 -0.0072138594 -1.172716e-01 0.036025938
## [12,] 0.0053790193 5.644747e-03 0.0085848889 2.712803e-02 -0.112521356
## [13,] 0.0044088426 6.421357e-04 0.0091056553 2.998261e-02 -0.005914511
## [14,] -0.0036521661 3.508191e-03 -0.0019546805 -1.747060e-03 -0.024746115
## [15,] -0.0548534911 -2.282676e-02 -0.0672228464 -3.791037e-02 -0.061596916
## [16,] -0.0010797275 -4.485426e-03 -0.0066370533 4.027293e-03 -0.024589776
## [17,] 0.0187974085 4.900733e-03 0.0215371637 1.805363e-02 -0.003517435
## [18,] 0.0044876299 -8.620030e-04 0.0055726992 4.284863e-03 0.022331037
## [19,] -0.0025084057 -5.059264e-03 -0.0061992115 4.062804e-04 0.005019732
## [20,] 0.0021582619 -3.170161e-03 -0.0014847718 -1.872899e-03 0.017629435
## [21,] -0.0037025730 -2.948543e-03 0.0012262941 3.825325e-04 0.023502515
## [22,] 0.0019632839 -5.134531e-04 0.0032256581 9.223576e-04 0.022937742
## [23,] -0.0272469315 -1.138932e-02 0.0124434683 1.405693e-03 0.027408322
## [24,] 0.0068827341 -4.906881e-03 0.0017671286 4.508565e-03 0.021717308
## [25,] 0.0086981338 7.571060e-02 -0.0029278292 -5.623598e-03 0.012960422
## [26,] 0.0022365789 -2.276930e-04 0.0449844652 -3.883827e-02 0.007469165
## [27,] 0.0117907055 1.497429e-03 -0.0358105154 -9.116900e-02 0.003328325
## [28,] 0.0195347584 6.257080e-03 -0.0033909017 7.589093e-03 -0.010948547
## [29,] 0.0049537801 8.239960e-03 -0.0004049058 4.798226e-02 -0.051741881
## [30,] 0.0270034586 1.244817e-02 0.0345797344 4.537257e-02 -0.031946419
## [31,] 0.0167813196 1.090848e-02 0.0251236150 1.397533e-02 0.025205358
## [32,] 0.0162055485 8.591566e-03 0.0225762354 8.958470e-03 0.028506100
## [33,] 0.0203467178 1.651588e-02 0.0351586117 2.017776e-02 0.037823492
## [34,] 0.0402636448 1.050900e-02 0.0454029999 2.230477e-02 0.041582152
## [35,] -0.0596553258 2.483705e-02 0.0348759245 2.302323e-02 0.044425830
## [36,] 0.6181032134 -3.485777e-02 0.0341874991 1.622305e-02 0.031738323
## [37,] -0.0348577720 7.835516e-01 -0.0503545770 1.772848e-02 0.023497756
## [38,] 0.0341874991 -5.035458e-02 0.7069703637 -1.642916e-01 -0.025377397
## [39,] 0.0162230539 1.772848e-02 -0.1642915565 6.875218e-01 -0.002923179
## [40,] 0.0317383229 2.349776e-02 -0.0253773968 -2.923179e-03 0.815607293
## [41,] -0.0062772000 8.765050e-04 0.0093037095 4.370735e-03 0.031578987
## [42,] 0.0137416280 7.660978e-03 0.0191864843 3.723719e-03 0.035166048
## [43,] -0.0069247166 1.341357e-03 -0.0023981827 -1.041435e-02 0.020414938
## [44,] -0.0043478509 5.078483e-03 0.0061066166 -2.620141e-03 0.025988008
## [45,] 0.0110540776 5.528948e-03 0.0139532031 -1.288390e-04 0.030327932
## [46,] 0.0209728381 1.659857e-03 0.0030050540 -4.218527e-03 0.020373240
## [47,] -0.0064661729 7.864083e-03 0.0010368248 -4.223771e-03 0.026530813
## [48,] 0.0853669241 2.946874e-02 -0.0043972661 -5.787776e-03 0.020783920
## [49,] 0.0251281592 5.823881e-02 -0.0410972349 -1.487145e-02 0.022176957
## [50,] -0.0124885235 1.105960e-02 0.0365069633 5.913660e-02 0.035949045
## [51,] -0.0030458401 -1.221582e-02 0.0329518578 -2.355858e-02 0.009704179
## [52,] -0.0008410007 -3.687256e-04 0.0205737817 -1.472512e-02 -0.003822303
## [53,] 0.0056693212 7.027481e-03 -0.0002725252 4.858314e-05 -0.062333437
## [54,] -0.0134370155 -4.279515e-03 -0.0160153759 -2.871788e-02 0.042041074
## [55,] -0.0123022908 -1.878563e-03 -0.0128855055 -1.116040e-02 0.006322271
## [56,] 0.0092020572 1.517526e-03 0.0034578773 -3.437439e-03 0.023088148
## [57,] -0.0194340532 -4.387914e-03 -0.0191297657 -1.652242e-02 0.005076973
## [58,] 0.0110086988 -1.233000e-03 -0.0072106187 -9.112227e-03 0.014349410
## [59,] 0.0004513626 -2.233664e-03 -0.0219192623 -1.719244e-02 0.005504620
## [60,] -0.2433973700 -3.242054e-02 0.0426191791 -9.412964e-03 0.011205533
## [61,] -0.0089757905 -1.877982e-01 -0.1145289030 -6.328998e-03 0.037070067
## [62,] -0.0058875871 -2.103440e-03 -0.1705212488 4.878364e-03 0.016026490
## [63,] -0.0111658616 8.596404e-03 0.0154147489 -1.457908e-01 0.045573809
## [64,] -0.0121435667 -1.073566e-03 0.0059229352 -7.913186e-03 -0.190475383
## [65,] 0.0030125445 -1.251775e-02 0.0168489770 -7.011706e-02 0.049607028
## [66,] 0.0014267339 -5.402290e-04 -0.0016107234 1.325900e-03 -0.003238096
## [,41] [,42] [,43] [,44] [,45]
## [1,] -0.0071563151 0.0049020592 -0.0100887688 -0.0023997818 -0.0007665168
## [2,] 0.0068305006 -0.0037605748 0.0014168834 -0.0064290815 0.0078584040
## [3,] 0.0026216197 0.0069187760 -0.0060747527 0.0126925609 -0.0002285053
## [4,] 0.0063395741 -0.0018876810 0.0061902876 0.0184628318 0.0014040000
## [5,] 0.0053641608 -0.0025442542 0.0058905711 -0.0027626719 0.0090978271
## [6,] 0.0041508928 0.0023588900 0.0107923213 0.0029369298 0.0088585200
## [7,] 0.0054436398 -0.0038182146 0.0031180765 -0.0138411381 0.0056422907
## [8,] 0.0049529797 0.0048398745 0.0112549928 0.0108639596 0.0061587659
## [9,] 0.0070004742 -0.0023487173 0.0091369241 0.0016205267 0.0026979428
## [10,] -0.0083791767 0.0045659435 0.0100106551 0.0057058968 0.0074821252
## [11,] 0.0256612638 -0.0138126159 0.0084581071 -0.0018403457 0.0009140833
## [12,] -0.0045018042 0.0225125065 0.0065538697 0.0107676836 0.0021293265
## [13,] -0.1973708965 -0.0084967589 0.0032959505 -0.0011685034 0.0045863864
## [14,] -0.0036783861 0.0229422006 0.0246423117 0.0148525847 0.0278850284
## [15,] 0.0131163041 -0.0377263264 0.0073630150 -0.0103746417 -0.0240583016
## [16,] 0.0023940657 -0.0392831733 -0.0610547475 -0.0493785320 -0.0583663073
## [17,] 0.0011179853 0.0042849727 -0.0271050894 -0.0172295521 -0.0110432953
## [18,] 0.0028571991 -0.1881755986 0.0178249914 -0.0066166265 -0.0264519577
## [19,] 0.0101627476 -0.0129189541 -0.1057689068 0.0061485308 -0.0094477197
## [20,] 0.0024071799 -0.0074191995 0.0350447889 -0.1990569694 0.0268457813
## [21,] 0.0031804235 -0.0203085896 -0.0224165809 0.0144632492 -0.1161064054
## [22,] 0.0019040978 -0.0123792830 -0.0115114011 -0.0060169846 0.0239490313
## [23,] 0.0022493200 -0.0099648974 -0.0128815509 0.0066027183 -0.0298602642
## [24,] 0.0023583189 -0.0139907556 -0.0167458254 -0.0032192523 -0.0198362859
## [25,] 0.0082140283 -0.0136775373 -0.0103490331 -0.0069999094 -0.0146233312
## [26,] -0.0116178887 -0.0040663301 -0.0023064948 0.0028604159 -0.0023132216
## [27,] 0.0281650620 -0.0145383873 -0.0025472251 0.0008563228 -0.0020198597
## [28,] -0.0588829307 0.0391820595 -0.0149110721 0.0025090579 -0.0031365303
## [29,] -0.1644545433 -0.0243602304 -0.0006268913 -0.0017684343 -0.0030199864
## [30,] 0.0360487396 -0.1611306876 0.0723154726 0.0060363454 0.0328502067
## [31,] -0.0037278812 -0.0333533923 0.0574846357 0.0397264926 -0.0002094054
## [32,] -0.0058303273 0.0175478879 0.0387729789 -0.1000291693 0.0371089727
## [33,] -0.0064714578 0.0278771256 0.0039997569 0.0602512395 -0.0541457073
## [34,] -0.0087249303 0.0255080630 -0.0088778947 0.0582120943 -0.0091270431
## [35,] -0.0060194816 0.0219887804 -0.0044334134 -0.0045916047 0.0164777512
## [36,] -0.0062772000 0.0137416280 -0.0069247166 -0.0043478509 0.0110540776
## [37,] 0.0008765050 0.0076609782 0.0013413575 0.0050784835 0.0055289481
## [38,] 0.0093037095 0.0191864843 -0.0023981827 0.0061066166 0.0139532031
## [39,] 0.0043707351 0.0037237187 -0.0104143491 -0.0026201415 -0.0001288390
## [40,] 0.0315789869 0.0351660479 0.0204149377 0.0259880075 0.0303279317
## [41,] 0.7906313811 -0.0090996537 -0.0004665397 -0.0015687401 -0.0036986728
## [42,] -0.0090996537 0.2611436653 -0.0237343739 0.0431533097 0.0376294570
## [43,] -0.0004665397 -0.0237343739 0.1273706383 0.0141832190 0.0690301272
## [44,] -0.0015687401 0.0431533097 0.0141832190 0.2504194803 -0.0298774832
## [45,] -0.0036986728 0.0376294570 0.0690301272 -0.0298774832 0.1381482709
## [46,] -0.0010236102 0.0315436258 0.0453534238 0.0659602260 0.0114090223
## [47,] -0.0009054076 0.0321238541 0.0604323202 0.0051097076 0.0844859682
## [48,] -0.0047280434 0.0286514371 0.0573956700 0.0434168008 0.0497805803
## [49,] 0.0201427622 0.0197008808 0.0398310827 0.0285668354 0.0349409605
## [50,] -0.0135540982 0.0089525113 0.0364337216 0.0270941234 0.0252371067
## [51,] 0.0172728401 0.0171769968 -0.0009707502 0.0034730694 -0.0020342586
## [52,] -0.0658959132 -0.0008440476 0.1029915521 0.0720677099 0.0841609601
## [53,] -0.1289853583 0.0179250010 -0.0290968669 -0.0143061125 -0.0206284128
## [54,] -0.0235751830 0.0907851819 -0.0040742152 0.0148474809 0.0043890186
## [55,] -0.0002979605 0.0058545660 0.0112927304 0.0113760376 0.0142110930
## [56,] -0.0061108599 0.0057620173 0.0026765830 0.0210063646 0.0370835509
## [57,] -0.0013911011 -0.0032543163 0.0277086756 0.0270820173 0.0177075765
## [58,] -0.0041415186 -0.0036339838 0.0224850654 -0.0633611767 0.0565605687
## [59,] -0.0008267653 0.0055247770 0.0319826975 0.0824263059 -0.0157441743
## [60,] -0.0151791423 0.0069055883 0.0275919752 0.0199044169 0.0203228302
## [61,] 0.0264466458 0.0151163238 0.0290769555 0.0267649221 0.0239946920
## [62,] -0.0111398923 0.0131689144 0.0363381157 0.0277184443 0.0277810204
## [63,] -0.0189216235 0.0197065409 0.0273681512 0.0212877672 0.0208865317
## [64,] 0.0438255216 -0.0088993423 0.0656012979 0.0461699033 0.0449482988
## [65,] -0.1553051621 0.0381927036 0.0311717239 0.0282117980 0.0332247909
## [66,] 0.0008584204 0.0015105475 -0.0022168498 -0.0004286292 -0.0014360714
## [,46] [,47] [,48] [,49] [,50]
## [1,] -0.005395759 -0.0020300767 -0.0092339567 -0.0035205649 -0.006783709
## [2,] 0.005003038 0.0017628199 0.0052731139 0.0010156825 -0.001460847
## [3,] 0.002905076 0.0060189298 0.0090204842 0.0032134074 0.004794522
## [4,] 0.007664360 -0.0067145124 0.0032810350 -0.0004241638 0.002922073
## [5,] -0.004236762 0.0142907578 0.0098012439 0.0031108232 0.002833716
## [6,] -0.016901841 0.0080227826 0.0041290069 0.0032943612 0.002470619
## [7,] 0.013764714 0.0143626319 0.0225004401 -0.0016330592 0.001563780
## [8,] 0.011164500 0.0016512116 -0.0201317686 -0.0186458855 -0.001696954
## [9,] 0.003030304 0.0049870147 -0.0100194254 0.0272998353 0.004229609
## [10,] 0.007340727 0.0029089920 0.0162104633 -0.0149281571 0.003223281
## [11,] 0.002016064 0.0022568087 0.0042752782 0.0104516890 0.006626123
## [12,] 0.008019441 0.0021294691 0.0089600506 0.0068504349 -0.003787212
## [13,] 0.001777548 0.0021385013 -0.0006985036 0.0040170280 -0.014230815
## [14,] 0.014616033 0.0180895855 0.0136578509 0.0098286963 -0.013956717
## [15,] -0.002747722 -0.0072691282 0.0073137285 0.0078062103 0.026834657
## [16,] -0.046530070 -0.0535882153 -0.0516204293 -0.0355968801 -0.028258867
## [17,] -0.017675602 -0.0200813334 -0.0258769180 -0.0211593528 -0.034025638
## [18,] -0.010844617 -0.0175145351 -0.0100341205 -0.0076975663 0.013270253
## [19,] -0.011947301 0.0055305877 -0.0024342443 -0.0004587154 0.018206844
## [20,] 0.004605268 -0.0219550007 -0.0037749758 -0.0039691978 0.013854893
## [21,] -0.024672818 0.0299041400 -0.0089357141 -0.0014176109 0.013658474
## [22,] -0.207527435 0.0418242894 -0.0080808362 -0.0128992478 0.016698192
## [23,] 0.055683160 -0.2160145831 0.0020133364 0.0387702596 0.004506044
## [24,] 0.016532852 0.0041047761 -0.1933540471 0.0272876984 0.017093321
## [25,] -0.012580567 -0.0096219537 0.0349096728 -0.1724487185 0.007365971
## [26,] 0.002289992 0.0002724940 0.0245682123 -0.0331357086 -0.202886953
## [27,] -0.002151278 -0.0010534224 -0.0160033611 0.0278007265 -0.078076279
## [28,] 0.002569589 0.0041811948 -0.0010642582 0.0208301705 0.004167632
## [29,] -0.004669085 -0.0038088680 -0.0061556046 0.0061747921 0.009241510
## [30,] 0.009476014 0.0228732917 0.0116456946 0.0054867057 -0.002614819
## [31,] 0.013474451 -0.0027339545 0.0054146579 0.0032136408 -0.005710802
## [32,] 0.014616915 0.0059869348 0.0087012249 0.0040152188 -0.001565441
## [33,] 0.015173151 -0.0306663465 0.0028419489 -0.0013017774 -0.010674306
## [34,] -0.004449759 0.0433687569 -0.0119020243 -0.0042483553 -0.014042789
## [35,] 0.022130199 0.0337406398 -0.0037179782 -0.0150233843 -0.018842030
## [36,] 0.020972838 -0.0064661729 0.0853669241 0.0251281592 -0.012488523
## [37,] 0.001659857 0.0078640833 0.0294687424 0.0582388073 0.011059602
## [38,] 0.003005054 0.0010368248 -0.0043972661 -0.0410972349 0.036506963
## [39,] -0.004218527 -0.0042237708 -0.0057877762 -0.0148714487 0.059136604
## [40,] 0.020373240 0.0265308126 0.0207839198 0.0221769574 0.035949045
## [41,] -0.001023610 -0.0009054076 -0.0047280434 0.0201427622 -0.013554098
## [42,] 0.031543626 0.0321238541 0.0286514371 0.0197008808 0.008952511
## [43,] 0.045353424 0.0604323202 0.0573956700 0.0398310827 0.036433722
## [44,] 0.065960226 0.0051097076 0.0434168008 0.0285668354 0.027094123
## [45,] 0.011409022 0.0844859682 0.0497805803 0.0349409605 0.025237107
## [46,] 0.358254624 -0.1496430285 0.0528287030 0.0225427336 0.028598854
## [47,] -0.149643029 0.3192109134 0.0257922841 0.0430258433 0.026417414
## [48,] 0.052828703 0.0257922841 0.3169376993 -0.1718341766 0.054321633
## [49,] 0.022542734 0.0430258433 -0.1718341766 0.6763679867 -0.057233221
## [50,] 0.028598854 0.0264174141 0.0543216330 -0.0572332206 0.654392923
## [51,] 0.003044453 0.0030400391 -0.0015649437 0.0159399615 -0.100481148
## [52,] 0.069205233 0.0798331217 0.0749351128 0.0595342768 0.012660339
## [53,] -0.015707972 -0.0166868093 -0.0160447958 -0.0046974743 0.016498077
## [54,] 0.012790048 0.0123520546 0.0171593610 0.0146180687 0.025321941
## [55,] 0.015965912 0.0202347404 0.0243536546 0.0198565649 0.030456931
## [56,] 0.011620482 0.0410360595 0.0187666463 0.0153273848 0.020498852
## [57,] 0.005799294 0.0228537670 0.0234969227 0.0205644760 0.032711982
## [58,] 0.137724060 -0.0414737485 0.0274631981 0.0133425920 0.026515856
## [59,] -0.067449718 0.0084577753 0.0309636715 0.0287076228 0.030702304
## [60,] 0.017520079 0.0238861747 -0.1034135125 0.0660075084 0.054924487
## [61,] 0.023609859 0.0315948896 0.0900521060 -0.1386480521 -0.069640701
## [62,] 0.025944248 0.0315353473 0.0261495255 0.0436520577 -0.243860946
## [63,] 0.019998748 0.0247256670 0.0251754821 0.0375424489 0.062864420
## [64,] 0.044093416 0.0513892009 0.0607915324 0.0250505875 0.074574283
## [65,] 0.027359132 0.0271852422 0.0371378136 -0.0280707594 0.017739497
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## [23,] 1.133392e-02 0.0010467682 1.228453e-02 0.0066852367 0.0053134203
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## [39,] -2.355858e-02 -0.0147251230 4.858314e-05 -0.0287178789 -0.0111604016
## [40,] 9.704179e-03 -0.0038223034 -6.233344e-02 0.0420410742 0.0063222705
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## [44,] 3.473069e-03 0.0720677099 -1.430611e-02 0.0148474809 0.0113760376
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## [46,] 3.044453e-03 0.0692052328 -1.570797e-02 0.0127900476 0.0159659124
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## [49,] 1.593996e-02 0.0595342768 -4.697474e-03 0.0146180687 0.0198565649
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## [51,] 6.167288e-01 0.1315017352 -8.079108e-02 0.0220611170 0.0248276976
## [52,] 1.315017e-01 0.4806103947 2.026615e-01 0.0097027832 0.0188968745
## [53,] -8.079108e-02 0.2026615426 6.443401e-01 0.0115060952 0.0134085116
## [54,] 2.206112e-02 0.0097027832 1.150610e-02 0.0850287392 0.0272957577
## [55,] 2.482770e-02 0.0188968745 1.340851e-02 0.0272957577 0.0902033697
## [56,] 1.667382e-02 0.0182102209 1.159299e-02 0.0240298499 0.0097499458
## [57,] 2.534128e-02 0.0214996837 1.078374e-02 0.0360792252 0.0466452760
## [58,] 1.947453e-02 0.0182529171 1.096996e-02 0.0266885368 0.0303327944
## [59,] 2.429633e-02 0.0302483480 7.855727e-03 0.0366160550 0.0495093840
## [60,] 2.269597e-02 0.0270865122 4.575756e-03 0.0274958340 0.0367268752
## [61,] 7.926264e-03 0.0380427295 1.529733e-02 0.0196685261 0.0280827648
## [62,] 7.085863e-02 0.0524859794 -1.093376e-02 0.0265726714 0.0359356647
## [63,] -2.436636e-01 -0.0268345120 4.729046e-02 0.0282504564 0.0186877575
## [64,] -5.330941e-02 -0.0805106732 -9.540665e-02 0.0296481579 0.0862206352
## [65,] -8.524590e-03 -0.0896374047 -2.768191e-01 0.0216933926 0.0046344088
## [66,] 1.606402e-03 -0.0021597783 -6.278649e-04 0.0004723154 -0.0012132284
## [,56] [,57] [,58] [,59] [,60]
## [1,] 0.0133772438 0.0030039096 0.0086846490 0.0028012852 0.009763095
## [2,] -0.0239183300 -0.0115227081 -0.0119222741 -0.0126097990 -0.014344408
## [3,] 0.0015896458 -0.0084600767 -0.0061551110 -0.0084595432 -0.009704835
## [4,] -0.0052163990 -0.0081464459 -0.0174082756 -0.0025721324 -0.012755921
## [5,] 0.0119448206 0.0070699698 0.0017583326 -0.0231338260 -0.010177820
## [6,] -0.0201079600 -0.0055884903 -0.0049292833 0.0118005917 -0.014893587
## [7,] 0.0019030889 -0.0202869102 0.0185307845 -0.0076325564 -0.008274553
## [8,] -0.0183788508 -0.0077163074 -0.0221427761 -0.0057542892 0.020352355
## [9,] -0.0119987824 -0.0073197534 -0.0072825735 -0.0089844422 0.005172829
## [10,] -0.0140776200 -0.0113459869 -0.0147321550 -0.0089625452 -0.016295731
## [11,] -0.0095886067 -0.0009126114 -0.0029667243 -0.0014376835 -0.007346019
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## [15,] 0.0018715350 0.0440417747 0.0221776199 0.0415662736 0.015578850
## [16,] -0.0149138075 -0.0168097884 -0.0139096394 -0.0249577610 -0.022324102
## [17,] -0.0287687695 -0.0495189257 -0.0372088617 -0.0501925430 -0.037991777
## [18,] 0.0165779603 0.0162943602 0.0099529411 0.0117158494 0.011260732
## [19,] 0.0220284103 0.0130501036 0.0341604678 -0.0049922760 0.010682809
## [20,] -0.0315618049 0.0207186801 0.0195172027 0.0013612345 0.006223554
## [21,] 0.0219066903 -0.0021542478 -0.0065757252 0.0093395539 0.009502452
## [22,] 0.0088412053 0.0137201121 -0.0432120886 0.0656450065 0.002585545
## [23,] -0.0028922886 0.0068514136 0.0120783811 0.0397884770 0.031462806
## [24,] 0.0090327713 0.0111463202 0.0191879069 -0.0006985245 0.037246044
## [25,] 0.0104892576 0.0145355131 0.0095008979 0.0107797611 -0.012569092
## [26,] 0.0103639422 0.0054380730 0.0107580603 -0.0014299856 0.011175159
## [27,] 0.0018058184 -0.0055586039 -0.0035121117 -0.0055745424 -0.011801786
## [28,] 0.0180973595 0.0111261284 0.0196120035 -0.0003620087 0.003673113
## [29,] -0.0032965339 -0.0092681686 -0.0077917067 -0.0073863110 -0.013043843
## [30,] 0.0136201360 -0.0169344433 0.0102696303 -0.0225637508 0.000797582
## [31,] -0.0040413081 -0.0019875815 -0.0451500487 0.0227605217 -0.001081674
## [32,] -0.2427381226 0.0204712934 -0.0180597694 0.0027094843 0.000717424
## [33,] -0.0534330167 -0.1302129716 -0.0315713937 0.0346806797 -0.004129133
## [34,] -0.0369794743 0.0344121973 -0.1937800008 0.0158360680 -0.009296833
## [35,] 0.0096030124 -0.0426987724 0.0435562481 -0.2003814410 0.016217311
## [36,] 0.0092020572 -0.0194340532 0.0110086988 0.0004513626 -0.243397370
## [37,] 0.0015175262 -0.0043879141 -0.0012329999 -0.0022336643 -0.032420540
## [38,] 0.0034578773 -0.0191297657 -0.0072106187 -0.0219192623 0.042619179
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## [40,] 0.0230881477 0.0050769729 0.0143494098 0.0055046198 0.011205533
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## [42,] 0.0057620173 -0.0032543163 -0.0036339838 0.0055247770 0.006905588
## [43,] 0.0026765830 0.0277086756 0.0224850654 0.0319826975 0.027591975
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## [45,] 0.0370835509 0.0177075765 0.0565605687 -0.0157441743 0.020322830
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## [47,] 0.0410360595 0.0228537670 -0.0414737485 0.0084577753 0.023886175
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## [49,] 0.0153273848 0.0205644760 0.0133425920 0.0287076228 0.066007508
## [50,] 0.0204988517 0.0327119817 0.0265158560 0.0307023039 0.054924487
## [51,] 0.0166738205 0.0253412836 0.0194745332 0.0242963297 0.022695973
## [52,] 0.0182102209 0.0214996837 0.0182529171 0.0302483480 0.027086512
## [53,] 0.0115929903 0.0107837419 0.0109699594 0.0078557273 0.004575756
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## [57,] 0.0016891977 0.0930744302 -0.0089448933 0.0746616531 0.036864391
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## [60,] 0.0288866311 0.0368643907 0.0315720935 0.0210434133 0.216196443
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## [62,] 0.0257939524 0.0369385209 0.0289823520 0.0412905709 0.028776885
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## [64,] 0.0595270804 0.0845595315 0.0675439879 0.0899122647 0.085553199
## [65,] 0.0083739845 0.0054863219 0.0068501128 0.0066022937 0.037034583
## [66,] -0.0007118702 -0.0006382275 -0.0007119330 -0.0014412263 -0.001437497
## [,61] [,62] [,63] [,64] [,65]
## [1,] 0.0104044142 5.728068e-03 -3.419157e-03 0.0209608083 0.0003527435
## [2,] -0.0115929036 -9.374377e-03 -8.943102e-03 -0.0085647920 -0.0113280199
## [3,] -0.0094977385 -6.571403e-03 6.241569e-03 -0.0323573398 0.0067236546
## [4,] -0.0118405742 -9.370131e-03 -1.684691e-03 -0.0239848394 -0.0026816631
## [5,] -0.0101525183 -7.886450e-03 1.929349e-03 -0.0266088419 0.0012888016
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## [7,] -0.0128007927 -9.636796e-03 -3.332741e-04 -0.0283055253 -0.0031413913
## [8,] 0.0151152760 -8.658827e-03 2.141076e-03 -0.0263821304 0.0025387744
## [9,] -0.0149840042 -1.366513e-03 -4.951968e-03 -0.0254267385 -0.0009411105
## [10,] -0.0028193023 -7.409708e-03 1.829574e-02 -0.0111549692 0.0163543524
## [11,] -0.0016943779 5.255333e-03 1.595621e-02 -0.0209826817 -0.0122243577
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## [13,] 0.0064127982 -1.735602e-03 -1.017475e-02 0.0099381145 0.0517188715
## [14,] 0.0043901875 7.247816e-03 5.292399e-03 0.0045922966 0.0104690932
## [15,] 0.0003724539 1.765083e-02 1.901350e-02 0.0347588717 -0.0104445371
## [16,] -0.0269419998 -3.115579e-02 -2.384871e-02 -0.0514272595 -0.0325452967
## [17,] -0.0275585569 -3.766240e-02 -2.247217e-02 -0.0852791175 -0.0057663702
## [18,] 0.0015576461 -1.210974e-03 -8.640609e-03 0.0248858203 -0.0169420809
## [19,] 0.0006178148 1.705025e-03 -2.399805e-03 0.0185187425 -0.0104712607
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## [22,] 0.0014793587 -2.520089e-03 -1.026163e-03 0.0036390938 -0.0031114287
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## [24,] -0.0089147907 -5.729733e-03 -1.601205e-03 0.0006104786 -0.0043066024
## [25,] 0.0305692711 1.200857e-02 -1.394701e-02 -0.0033836824 -0.0116242270
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## [27,] 0.0293778669 5.752774e-02 5.152550e-02 -0.0030166177 -0.0111205966
## [28,] 0.0284116775 1.574094e-02 3.741232e-03 0.0263098612 -0.0001738525
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## [39,] -0.0063289980 4.878364e-03 -1.457908e-01 -0.0079131859 -0.0701170645
## [40,] 0.0370700674 1.602649e-02 4.557381e-02 -0.1904753829 0.0496070281
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## [43,] 0.0290769555 3.633812e-02 2.736815e-02 0.0656012979 0.0311717239
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## [45,] 0.0239946920 2.778102e-02 2.088653e-02 0.0449482988 0.0332247909
## [46,] 0.0236098586 2.594425e-02 1.999875e-02 0.0440934156 0.0273591321
## [47,] 0.0315948896 3.153535e-02 2.472567e-02 0.0513892009 0.0271852422
## [48,] 0.0900521060 2.614953e-02 2.517548e-02 0.0607915324 0.0371378136
## [49,] -0.1386480521 4.365206e-02 3.754245e-02 0.0250505875 -0.0280707594
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## [51,] 0.0079262635 7.085863e-02 -2.436636e-01 -0.0533094141 -0.0085245905
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## [53,] 0.0152973263 -1.093376e-02 4.729046e-02 -0.0954066549 -0.2768190722
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## [55,] 0.0280827648 3.593566e-02 1.868776e-02 0.0862206352 0.0046344088
## [56,] 0.0210039519 2.579395e-02 1.282297e-02 0.0595270804 0.0083739845
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## [58,] 0.0225737814 2.898235e-02 1.592893e-02 0.0675439879 0.0068501128
## [59,] 0.0378445120 4.129057e-02 2.526320e-02 0.0899122647 0.0066022937
## [60,] -0.0849586665 2.877689e-02 3.021535e-03 0.0855531992 0.0370345834
## [61,] 0.4418746921 4.445450e-02 5.836922e-02 0.0357137049 -0.0649127298
## [62,] 0.0444545002 2.133375e-01 -2.879642e-02 0.0718278618 0.0397608647
## [63,] 0.0583692175 -2.879642e-02 1.794697e-01 0.0511025888 0.0086370152
## [64,] 0.0357137049 7.182786e-02 5.110259e-02 0.3559934776 0.0913892586
## [65,] -0.0649127298 3.976086e-02 8.637015e-03 0.0913892586 0.2765607019
## [66,] 0.0001890885 5.190561e-05 -1.735812e-03 -0.0020379262 -0.0008997256
## [,66]
## [1,] 1.158441e-03
## [2,] -1.973227e-04
## [3,] -1.527481e-03
## [4,] -4.290852e-04
## [5,] -1.951885e-03
## [6,] -1.736935e-03
## [7,] -1.938593e-03
## [8,] -5.927517e-04
## [9,] -1.644957e-03
## [10,] -3.278450e-04
## [11,] -2.822354e-03
## [12,] -1.529455e-03
## [13,] 1.338728e-03
## [14,] 9.941847e-04
## [15,] 6.513935e-04
## [16,] 1.170427e-03
## [17,] 7.823946e-04
## [18,] -3.350941e-03
## [19,] 4.903678e-04
## [20,] -1.845255e-04
## [21,] -2.066360e-03
## [22,] -2.290041e-03
## [23,] -3.596957e-03
## [24,] 1.722101e-03
## [25,] 2.471111e-03
## [26,] -6.296649e-03
## [27,] 2.228085e-03
## [28,] -6.468791e-04
## [29,] -2.058077e-03
## [30,] -4.136589e-03
## [31,] 1.297019e-03
## [32,] -2.637253e-03
## [33,] -8.377401e-05
## [34,] 2.258945e-04
## [35,] 7.066990e-04
## [36,] 1.426734e-03
## [37,] -5.402290e-04
## [38,] -1.610723e-03
## [39,] 1.325900e-03
## [40,] -3.238096e-03
## [41,] 8.584204e-04
## [42,] 1.510547e-03
## [43,] -2.216850e-03
## [44,] -4.286292e-04
## [45,] -1.436071e-03
## [46,] 1.003144e-03
## [47,] -1.107623e-03
## [48,] -3.393870e-03
## [49,] -1.790917e-04
## [50,] -1.943251e-03
## [51,] 1.606402e-03
## [52,] -2.159778e-03
## [53,] -6.278649e-04
## [54,] 4.723154e-04
## [55,] -1.213228e-03
## [56,] -7.118702e-04
## [57,] -6.382275e-04
## [58,] -7.119330e-04
## [59,] -1.441226e-03
## [60,] -1.437497e-03
## [61,] 1.890885e-04
## [62,] 5.190561e-05
## [63,] -1.735812e-03
## [64,] -2.037926e-03
## [65,] -8.997256e-04
## [66,] 5.342066e-03
##
## $log_evidence
## [1] -160.9416
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 136
Use the viz_post_coefs() function to visualize
the posterior coefficient summaries for model 3 and model 6, based on
the strong prior specification.
post_means_3_1 <- laplace_03_strong$mode[1:ncol(X03)]
post_sds_3_1 <- sqrt(diag(laplace_03_strong$var_matrix)[1:ncol(X03)])
viz_post_coefs(post_means_3_1, post_sds_3_1, xnames_3)
post_means_6_1 <- laplace_06_strong$mode[1:ncol(X06)]
post_sds_6_1 <- sqrt(diag(laplace_06_strong$var_matrix)[1:ncol(X06)])
viz_post_coefs(post_means_6_1, post_sds_6_1, xnames_6)
### 3h)
You will fit one more set of Bayesian models with a very strong prior on the regression coefficients. The prior standard deviation will be equal to 1/50.
Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the very strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.
Run the Laplace Approximation using the strong prior for both
the model 3 and model 6 formulations. Assign the results to
laplace_03_very_strong and
laplace_06_very_strong.
Confirm that the optimizations converged for both laplace approximation results.
info_03_very_strong <- list(
yobs = df$y ,
design_matrix = X03,
mu_beta = 0,
tau_beta = 1/50,
sigma_rate = 1
)
info_06_very_strong <- list(
yobs = df$y,
design_matrix = X06,
mu_beta = 0,
tau_beta = 1/50,
sigma_rate = 1
)
Execute the Laplace Approximation.
laplace_03_very_strong <- my_laplace(rep(0, ncol(X03)+1), lm_logpost, info_03_very_strong)
laplace_03_very_strong
## $mode
## [1] 0.003592816 0.006873373 -0.003563880 -0.005548824 -0.034591622
## [6] -0.005597032 0.006696299 -0.010922109 -0.036247874 -0.082665512
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 3.845484e-04 2.590441e-07 -1.379062e-05 -2.342354e-07 -1.718063e-05
## [2,] 2.590441e-07 3.840058e-04 3.516810e-06 -1.839586e-06 -3.016304e-06
## [3,] -1.379062e-05 3.516810e-06 3.644930e-04 -1.862476e-06 -1.159400e-05
## [4,] -2.342354e-07 -1.839586e-06 -1.862476e-06 3.837901e-04 2.426885e-06
## [5,] -1.718063e-05 -3.016304e-06 -1.159400e-05 2.426885e-06 3.747607e-04
## [6,] -9.397066e-07 -3.102059e-06 -3.791049e-06 2.222883e-06 -5.962894e-06
## [7,] 1.343445e-06 -1.574856e-05 6.109597e-06 -9.945576e-06 2.864430e-06
## [8,] -3.066294e-06 -6.377982e-06 -2.351642e-06 -1.551118e-05 6.543867e-07
## [9,] -1.471934e-05 2.996232e-06 -3.702898e-05 -2.145648e-06 -2.186025e-05
## [10,] -6.498831e-05 -6.513476e-05 -1.287704e-05 5.111107e-05 3.033540e-04
## [,6] [,7] [,8] [,9] [,10]
## [1,] -9.397066e-07 1.343445e-06 -3.066294e-06 -1.471934e-05 -6.498831e-05
## [2,] -3.102059e-06 -1.574856e-05 -6.377982e-06 2.996232e-06 -6.513476e-05
## [3,] -3.791049e-06 6.109597e-06 -2.351642e-06 -3.702898e-05 -1.287704e-05
## [4,] 2.222883e-06 -9.945576e-06 -1.551118e-05 -2.145648e-06 5.111107e-05
## [5,] -5.962894e-06 2.864430e-06 6.543867e-07 -2.186025e-05 3.033540e-04
## [6,] 3.826461e-04 -6.989646e-06 9.383095e-06 -1.666291e-05 3.746312e-05
## [7,] -6.989646e-06 3.529481e-04 -2.161345e-05 8.768167e-06 -5.358861e-05
## [8,] 9.383095e-06 -2.161345e-05 3.545232e-04 2.842754e-06 1.034925e-04
## [9,] -1.666291e-05 8.768167e-06 2.842754e-06 3.207897e-04 2.649719e-04
## [10,] 3.746312e-05 -5.358861e-05 1.034925e-04 2.649719e-04 5.546765e-03
##
## $log_evidence
## [1] -140.2491
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 76
laplace_06_very_strong <- my_laplace(rep(0, ncol(X06)+1), lm_logpost, info_06_very_strong)
laplace_06_very_strong
## $mode
## [1] 9.596943e-03 2.025636e-04 4.495974e-04 1.246452e-03 2.046718e-03
## [6] 1.146167e-03 2.338609e-04 1.993863e-03 1.798106e-03 9.420842e-04
## [11] 1.640120e-03 -8.309383e-05 2.354423e-04 -7.060179e-03 -1.630535e-02
## [16] -1.278185e-02 -5.099149e-02 -8.855226e-04 9.591070e-04 -1.883520e-04
## [21] -4.065259e-04 2.894725e-04 -2.912267e-04 -1.957848e-03 -1.221818e-03
## [26] -1.863038e-04 5.372941e-05 -1.753379e-03 3.442692e-04 -8.925697e-05
## [31] -4.985172e-04 -1.637362e-04 -3.140736e-03 -2.410259e-03 -1.819216e-03
## [36] -7.196701e-04 3.763246e-05 -5.751050e-04 -2.466661e-04 -2.427530e-03
## [41] 3.903263e-04 -3.534247e-03 3.819679e-03 -3.536720e-04 8.327978e-04
## [46] 1.454220e-03 -1.924209e-03 -2.628542e-03 -1.208920e-03 -1.183356e-03
## [51] -7.888923e-04 -4.438683e-03 7.272193e-04 5.321576e-03 6.989395e-04
## [56] -9.418775e-04 -1.477772e-02 -1.059962e-02 -5.658545e-03 -3.733996e-03
## [61] -1.390625e-03 -2.460287e-03 -2.553944e-03 -6.012999e-03 1.946399e-04
## [66] -1.902831e-01
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 3.837139e-04 -1.519321e-06 -1.564882e-06 -1.076749e-06 -1.257282e-06
## [2,] -1.519321e-06 3.991731e-04 -4.445632e-07 -1.235545e-08 5.999702e-08
## [3,] -1.564882e-06 -4.445632e-07 3.989231e-04 -3.276401e-07 2.571449e-08
## [4,] -1.076749e-06 -1.235545e-08 -3.276401e-07 3.993237e-04 -3.817928e-07
## [5,] -1.257282e-06 5.999702e-08 2.571449e-08 -3.817928e-07 3.989649e-04
## [6,] -1.109776e-06 4.152320e-08 4.574653e-08 4.069365e-08 -3.592712e-07
## [7,] -1.250205e-06 2.339550e-08 2.508779e-08 2.675182e-08 3.756982e-09
## [8,] -1.834348e-06 3.848942e-08 4.994831e-08 7.862574e-08 1.358519e-07
## [9,] -1.249469e-06 2.654528e-08 3.666408e-08 6.377212e-08 1.107183e-07
## [10,] -1.207298e-06 3.377763e-08 4.033826e-08 5.276396e-08 8.857227e-08
## [11,] -2.984418e-07 1.809489e-07 4.784079e-08 6.885909e-08 1.173619e-07
## [12,] -1.676746e-06 -3.308902e-07 1.674387e-08 2.450361e-08 3.775493e-08
## [13,] 3.838930e-07 2.073298e-07 5.204072e-09 2.884237e-09 5.113285e-09
## [14,] -8.550244e-07 -6.832992e-08 2.559590e-07 -3.131557e-07 -8.770412e-08
## [15,] -1.289708e-05 -7.655103e-07 -9.778822e-07 -9.142998e-07 -1.632681e-06
## [16,] -5.381599e-06 3.676965e-07 -2.667658e-07 -4.497462e-07 3.740820e-07
## [17,] -1.674575e-05 -1.778038e-06 -1.544202e-06 -1.242393e-06 -2.595720e-06
## [18,] -5.750375e-07 -6.388654e-08 2.977858e-08 -4.156419e-08 -1.103108e-07
## [19,] 6.691208e-07 1.101250e-07 2.470961e-07 3.684964e-08 3.930172e-08
## [20,] -1.667181e-07 8.788874e-09 4.976082e-10 -1.320205e-07 -7.651909e-08
## [21,] -1.576359e-07 -4.657183e-08 -4.760074e-08 -1.065255e-07 6.655278e-08
## [22,] 2.135272e-07 -2.217752e-08 -1.357117e-08 -7.481108e-09 1.454594e-07
## [23,] -1.467427e-07 6.153365e-09 7.647601e-09 5.037906e-09 1.330210e-08
## [24,] 1.796292e-07 -8.322053e-09 -1.707008e-08 -4.730141e-08 -8.929771e-08
## [25,] 2.113376e-07 -6.420554e-09 -1.238423e-08 -3.168123e-08 -5.813182e-08
## [26,] 2.305971e-07 3.659493e-08 3.794353e-08 3.333505e-08 5.261776e-08
## [27,] 1.416995e-07 7.252342e-08 4.696737e-08 4.327278e-08 7.285785e-08
## [28,] -8.428031e-07 -6.210066e-08 -1.041957e-09 -2.613757e-08 -5.768223e-08
## [29,] 6.629177e-08 4.842842e-08 1.018248e-08 1.242769e-08 2.509570e-08
## [30,] -2.068694e-08 -6.364691e-07 -1.689326e-07 7.498691e-08 2.557167e-07
## [31,] -6.934362e-07 -2.509855e-07 -9.715332e-07 -2.225146e-07 1.284143e-07
## [32,] -6.556152e-07 1.371652e-08 -2.455642e-07 -4.316792e-07 -3.438460e-07
## [33,] -1.680073e-06 1.197399e-07 7.260871e-08 -4.056919e-07 -1.265041e-06
## [34,] -1.648618e-06 7.303616e-08 5.210023e-08 -1.049083e-08 -4.872420e-07
## [35,] -1.489113e-06 2.992818e-08 2.126823e-08 -1.251886e-08 -5.861654e-08
## [36,] -1.485716e-06 2.933567e-08 2.715551e-08 1.060371e-08 1.463767e-08
## [37,] -6.111615e-07 1.764508e-08 1.868058e-08 1.694841e-08 2.824581e-08
## [38,] -4.960276e-07 7.918334e-08 8.092667e-08 6.500020e-08 1.036678e-07
## [39,] -6.098736e-08 1.847208e-07 7.662625e-08 6.637148e-08 1.045900e-07
## [40,] -1.422680e-06 -2.115330e-07 4.941677e-08 5.765748e-09 4.617188e-09
## [41,] 2.122005e-07 1.470532e-07 -9.390409e-09 -3.158401e-09 -8.084341e-09
## [42,] -3.385838e-06 2.224558e-07 -3.590062e-07 -3.513994e-07 -7.247000e-07
## [43,] 1.596640e-06 2.573600e-07 6.565460e-08 8.925257e-08 1.813037e-07
## [44,] -6.452458e-08 -3.003364e-09 -5.889629e-08 -1.107412e-07 -6.305295e-09
## [45,] -3.836549e-08 -1.882791e-07 -1.554138e-07 -9.362926e-08 4.525258e-07
## [46,] 5.649393e-07 -8.476549e-08 -5.263209e-08 -3.404431e-08 5.003071e-07
## [47,] -5.661797e-07 1.431463e-08 1.170743e-08 -1.762456e-08 -1.592168e-08
## [48,] -4.665317e-07 1.326987e-08 3.609210e-09 -3.951104e-08 -8.094951e-08
## [49,] -9.887992e-08 1.077033e-08 6.296524e-09 -1.425080e-08 -3.045701e-08
## [50,] -7.369209e-08 1.558924e-07 1.592102e-07 1.286692e-07 2.047467e-07
## [51,] 2.071709e-07 2.228181e-07 1.949602e-07 1.626013e-07 2.738781e-07
## [52,] -2.038103e-06 -7.384519e-08 -9.526350e-09 -7.110338e-08 -1.640892e-07
## [53,] 3.406853e-07 8.504126e-08 4.122186e-08 4.218652e-08 8.736454e-08
## [54,] 6.654122e-06 -2.006310e-06 3.074057e-07 8.614761e-07 2.060453e-06
## [55,] 5.877754e-07 -6.771396e-07 -2.749600e-06 -3.449398e-07 8.684285e-07
## [56,] -1.225256e-06 7.376936e-08 -6.448106e-07 -9.709896e-07 -8.948254e-07
## [57,] -5.018402e-06 4.355426e-07 2.710203e-07 -1.173946e-06 -3.798048e-06
## [58,] -4.221252e-06 2.368078e-07 1.471119e-07 -8.083826e-08 -1.500448e-06
## [59,] -2.988899e-06 7.226641e-08 4.569579e-08 -5.569023e-08 -1.748225e-07
## [60,] -2.408519e-06 5.283806e-08 3.813737e-08 -3.149796e-08 -6.805998e-08
## [61,] -9.087086e-07 3.642180e-08 3.189334e-08 4.311014e-09 1.152071e-09
## [62,] -4.699933e-07 3.232340e-07 3.296436e-07 2.646690e-07 4.230983e-07
## [63,] -3.936609e-07 5.705443e-07 2.722229e-07 2.192270e-07 3.274238e-07
## [64,] -1.870912e-06 -4.238568e-07 2.946487e-07 1.478002e-07 2.671780e-07
## [65,] -2.623988e-07 3.282216e-07 -1.039126e-07 -8.065800e-08 -1.634222e-07
## [66,] -1.260574e-04 -3.626361e-06 -6.311082e-06 -1.493025e-05 -2.623247e-05
## [,6] [,7] [,8] [,9] [,10]
## [1,] -1.109776e-06 -1.250205e-06 -1.834348e-06 -1.249469e-06 -1.207298e-06
## [2,] 4.152320e-08 2.339550e-08 3.848942e-08 2.654528e-08 3.377763e-08
## [3,] 4.574653e-08 2.508779e-08 4.994831e-08 3.666408e-08 4.033826e-08
## [4,] 4.069365e-08 2.675182e-08 7.862574e-08 6.377212e-08 5.276396e-08
## [5,] -3.592712e-07 3.756982e-09 1.358519e-07 1.107183e-07 8.857227e-08
## [6,] 3.992701e-04 -3.652864e-07 2.221968e-08 6.800429e-08 5.642474e-08
## [7,] -3.652864e-07 3.993190e-04 -3.690865e-07 -1.907561e-09 2.952867e-08
## [8,] 2.221968e-08 -3.690865e-07 3.986083e-04 -5.015527e-07 3.593506e-08
## [9,] 6.800429e-08 -1.907561e-09 -5.015527e-07 3.992333e-04 -2.951211e-07
## [10,] 5.642474e-08 2.952867e-08 3.593506e-08 -2.951211e-07 3.992292e-04
## [11,] 7.153409e-08 3.075759e-08 9.561291e-08 5.273807e-08 -3.729151e-07
## [12,] 2.740392e-08 2.265035e-08 2.862501e-08 9.975837e-09 -1.258796e-07
## [13,] 1.895879e-09 -1.452365e-09 5.529101e-09 1.087090e-08 4.180457e-08
## [14,] 9.153257e-08 -2.122896e-07 3.381300e-07 2.605445e-07 -1.477257e-07
## [15,] -1.413540e-06 -1.072335e-06 -1.547007e-06 -8.666124e-07 -8.540470e-07
## [16,] 2.950458e-07 -5.040960e-07 -6.186057e-07 -3.995322e-07 -1.520142e-06
## [17,] -1.698675e-06 -8.595517e-07 -1.176875e-06 -7.766387e-07 -1.597907e-06
## [18,] -7.183237e-08 -1.870980e-08 -5.463448e-08 -4.328024e-08 -3.028541e-08
## [19,] 2.480451e-08 1.741412e-08 5.040355e-08 4.201761e-08 5.072285e-08
## [20,] -8.415158e-09 4.808187e-09 -3.180114e-09 -4.449448e-09 1.088968e-08
## [21,] 1.106016e-07 -9.773000e-09 -4.770929e-08 -3.686993e-08 -1.901321e-08
## [22,] 1.734109e-07 -4.358655e-08 -2.322016e-08 -2.404004e-10 6.596301e-09
## [23,] -3.273093e-08 -1.737268e-07 -4.131356e-08 5.624809e-09 1.587341e-08
## [24,] -7.154553e-08 -5.151375e-08 3.156510e-07 1.411342e-07 -2.242782e-08
## [25,] -3.400915e-08 7.635810e-11 1.599008e-07 1.585759e-07 2.066964e-08
## [26,] 3.557977e-08 2.886154e-08 4.143373e-08 6.363559e-08 -5.299725e-08
## [27,] 4.845908e-08 3.255468e-08 4.749011e-08 3.421820e-08 -6.824987e-08
## [28,] -3.076281e-08 7.841754e-09 -4.677624e-08 -4.647118e-08 -4.843687e-08
## [29,] 1.558277e-08 3.937542e-09 1.675159e-08 1.419238e-08 1.731877e-08
## [30,] 1.755555e-07 8.259817e-08 1.043146e-07 6.302210e-08 1.144692e-07
## [31,] 9.985607e-08 6.482915e-08 7.050605e-08 3.883455e-08 9.726996e-08
## [32,] 4.576997e-08 3.702257e-08 4.912109e-08 3.105309e-08 5.466462e-08
## [33,] -4.400859e-07 2.057449e-08 -8.131006e-09 -3.027241e-08 3.106650e-08
## [34,] -8.387073e-07 -4.895062e-07 -8.284154e-08 -3.213951e-08 1.348251e-08
## [35,] -5.307712e-07 -7.173043e-07 -2.813866e-07 -5.765654e-08 6.769729e-09
## [36,] -4.936159e-08 -2.293525e-07 -1.044730e-06 -3.582164e-07 1.730598e-08
## [37,] 2.035787e-08 -6.138817e-09 -3.356500e-07 -2.737345e-07 -1.291919e-07
## [38,] 7.239620e-08 6.055110e-08 5.929561e-08 -1.117459e-07 -6.463108e-07
## [39,] 7.057854e-08 5.704087e-08 7.272529e-08 2.984311e-08 -3.100082e-07
## [40,] 1.612564e-08 3.951713e-08 -2.460434e-08 -4.478629e-08 -1.050419e-07
## [41,] -7.937527e-09 -6.724786e-09 3.538177e-09 1.028622e-08 4.032289e-08
## [42,] -4.829355e-07 -1.604540e-07 -3.245248e-07 -2.370370e-07 -2.383910e-07
## [43,] 9.467207e-08 7.353569e-08 2.116831e-07 1.751692e-07 2.094606e-07
## [44,] -3.096916e-08 1.471957e-08 -3.656012e-09 -8.063586e-09 3.966682e-08
## [45,] 3.854712e-07 -6.172124e-08 -1.075114e-07 -6.924746e-08 -5.031207e-08
## [46,] 4.610482e-07 -1.600412e-07 -4.380679e-08 7.612283e-09 1.454454e-08
## [47,] -1.400024e-07 -3.627030e-07 -1.513211e-07 -5.047498e-08 1.705049e-08
## [48,] -8.002421e-08 -9.990984e-08 -2.459550e-07 -5.083184e-08 -1.867701e-09
## [49,] -1.440463e-08 2.731475e-09 -6.941742e-09 1.895353e-08 -1.179737e-07
## [50,] 1.419161e-07 1.190931e-07 1.341218e-07 -4.287190e-08 -8.236324e-07
## [51,] 1.871972e-07 1.352034e-07 1.640622e-07 8.424377e-08 -3.738133e-07
## [52,] -9.120650e-08 2.135625e-08 -1.126768e-07 -1.189516e-07 -2.045166e-07
## [53,] 5.657706e-08 1.691866e-08 4.593994e-08 4.080069e-08 9.051308e-08
## [54,] 1.382405e-06 5.707265e-07 9.098340e-07 6.159386e-07 8.508736e-07
## [55,] 5.905910e-07 3.371202e-07 4.635837e-07 2.966482e-07 5.511678e-07
## [56,] 1.715754e-07 1.270496e-07 1.555114e-07 9.432672e-08 1.873502e-07
## [57,] -1.273372e-06 1.189811e-07 -1.687243e-07 -2.337995e-07 2.206129e-08
## [58,] -1.992729e-06 -1.002391e-06 -2.957081e-07 -2.035277e-07 -1.915303e-08
## [59,] -1.150191e-06 -1.396940e-06 -4.667180e-07 -1.623801e-07 3.300283e-09
## [60,] -1.513708e-07 -3.140392e-07 -1.624907e-06 -5.619991e-07 1.722302e-08
## [61,] 1.002863e-08 -4.275814e-09 -4.882826e-07 -3.025212e-07 -2.901002e-07
## [62,] 2.939104e-07 2.455000e-07 2.660183e-07 -1.575100e-07 -1.871614e-06
## [63,] 2.271826e-07 2.191335e-07 2.300484e-07 9.387797e-08 -9.293262e-07
## [64,] 2.132891e-07 1.846603e-07 5.536494e-08 -3.538134e-08 -2.251565e-07
## [65,] -1.137247e-07 -5.225543e-08 -6.269617e-08 -2.644769e-08 7.304006e-08
## [66,] -1.526203e-05 -4.300049e-06 -2.218876e-05 -1.939114e-05 -1.241618e-05
## [,11] [,12] [,13] [,14] [,15]
## [1,] -2.984418e-07 -1.676746e-06 3.838930e-07 -8.550244e-07 -1.289708e-05
## [2,] 1.809489e-07 -3.308902e-07 2.073298e-07 -6.832992e-08 -7.655103e-07
## [3,] 4.784079e-08 1.674387e-08 5.204072e-09 2.559590e-07 -9.778822e-07
## [4,] 6.885909e-08 2.450361e-08 2.884237e-09 -3.131557e-07 -9.142998e-07
## [5,] 1.173619e-07 3.775493e-08 5.113285e-09 -8.770412e-08 -1.632681e-06
## [6,] 7.153409e-08 2.740392e-08 1.895879e-09 9.153257e-08 -1.413540e-06
## [7,] 3.075759e-08 2.265035e-08 -1.452365e-09 -2.122896e-07 -1.072335e-06
## [8,] 9.561291e-08 2.862501e-08 5.529101e-09 3.381300e-07 -1.547007e-06
## [9,] 5.273807e-08 9.975837e-09 1.087090e-08 2.605445e-07 -8.666124e-07
## [10,] -3.729151e-07 -1.258796e-07 4.180457e-08 -1.477257e-07 -8.540470e-07
## [11,] 3.993313e-04 -7.434666e-08 -1.333018e-07 -4.034315e-07 -6.337537e-07
## [12,] -7.434666e-08 3.993618e-04 1.350747e-07 -5.447775e-07 -1.025171e-06
## [13,] -1.333018e-07 1.350747e-07 3.996268e-04 -6.693949e-08 1.364972e-07
## [14,] -4.034315e-07 -5.447775e-07 -6.693949e-08 3.854277e-04 -5.797234e-07
## [15,] -6.337537e-07 -1.025171e-06 1.364972e-07 -5.797234e-07 3.801979e-04
## [16,] -1.686009e-06 -1.190856e-06 -6.151977e-08 -3.379213e-05 -6.456675e-06
## [17,] -1.716648e-06 -1.243791e-06 1.264394e-07 -8.805917e-07 -5.313877e-05
## [18,] -7.172254e-09 -8.914043e-08 4.791025e-08 -1.161310e-06 -3.585316e-07
## [19,] 7.060838e-08 2.431847e-08 6.511705e-09 -1.492399e-06 2.140465e-07
## [20,] 9.369027e-09 1.452754e-08 -7.452063e-10 -9.849688e-07 5.645669e-10
## [21,] -2.483240e-08 -3.007519e-09 1.541119e-09 -1.760698e-06 6.145698e-07
## [22,] 1.242561e-08 3.620781e-09 3.570675e-09 -1.607377e-06 4.636755e-07
## [23,] 1.404120e-08 1.926995e-08 -1.462228e-09 -1.256562e-06 -2.052442e-07
## [24,] -5.485238e-08 9.563913e-09 -8.460010e-09 -1.253758e-06 3.236918e-07
## [25,] -3.564940e-08 3.818931e-09 -5.153329e-09 -5.729760e-07 2.776068e-07
## [26,] -7.381944e-08 -1.410182e-09 6.167946e-09 -9.838415e-07 -6.433824e-07
## [27,] -2.595183e-07 -9.924655e-09 -5.159390e-08 -6.017925e-07 -6.803010e-07
## [28,] -8.238001e-08 -2.452305e-07 -8.279749e-09 -1.196349e-06 -7.070486e-07
## [29,] -3.672564e-08 1.560409e-09 -2.167409e-07 2.077234e-07 -5.367791e-08
## [30,] 2.172762e-07 -1.628832e-07 1.373135e-07 1.847011e-07 -8.150956e-07
## [31,] 9.170936e-08 8.584167e-08 -8.496671e-09 3.319922e-07 -2.023334e-06
## [32,] 5.261459e-08 5.004448e-08 -6.533962e-09 -1.234361e-07 -1.640479e-06
## [33,] -1.077104e-08 7.234099e-08 -2.455746e-08 1.134604e-06 -2.840721e-06
## [34,] -2.093552e-08 4.963683e-08 -1.812585e-08 9.067545e-07 -2.254085e-06
## [35,] -2.034800e-08 3.901713e-08 -1.212006e-08 -2.170200e-07 -1.525789e-06
## [36,] 1.180964e-08 3.563863e-08 -6.958137e-09 -2.394025e-08 -1.467711e-06
## [37,] 4.711290e-09 1.416982e-08 2.325194e-09 2.816994e-08 -6.802049e-07
## [38,] -3.130595e-07 -4.771561e-08 3.825772e-08 -1.063770e-06 -1.577589e-06
## [39,] -6.022644e-07 -3.206388e-08 -1.325704e-07 -1.152608e-06 -1.307899e-06
## [40,] -1.179901e-07 -5.098472e-07 8.263262e-08 -9.644929e-07 -1.631372e-06
## [41,] -1.281785e-07 8.838714e-08 -3.317733e-07 -9.968195e-08 2.001125e-07
## [42,] -2.169825e-07 -2.793952e-07 1.046587e-07 -3.255198e-06 -1.537808e-06
## [43,] 2.771580e-07 1.253527e-07 1.282627e-08 -4.419128e-06 -5.734921e-07
## [44,] 3.816294e-08 4.646769e-08 -1.531131e-09 -2.442973e-06 1.359063e-07
## [45,] -4.114458e-08 -3.526498e-08 1.801878e-08 -4.939579e-06 2.612450e-06
## [46,] 3.726954e-08 -2.142613e-09 1.423179e-08 -3.718307e-06 1.498786e-06
## [47,] -5.625748e-09 4.928918e-08 -9.863621e-09 -2.141715e-06 -5.134441e-07
## [48,] -4.042888e-08 4.160931e-08 -1.364946e-08 -1.685941e-06 -5.009010e-07
## [49,] -3.165965e-08 1.876286e-08 -1.443367e-09 -7.354414e-07 -2.441658e-07
## [50,] -4.032367e-07 -2.049082e-08 5.189167e-08 -2.509648e-06 -2.717195e-06
## [51,] -8.619167e-07 -1.276665e-07 -1.785099e-07 -2.035078e-06 -2.655996e-06
## [52,] -3.840850e-07 -5.534860e-07 1.192660e-08 -2.645357e-06 -2.160539e-06
## [53,] -1.328912e-07 4.022511e-08 -3.523298e-07 4.758640e-07 2.334192e-08
## [54,] 1.154484e-06 1.665468e-08 2.678160e-07 3.137169e-06 -4.029583e-07
## [55,] 5.755211e-07 4.311271e-07 -3.118368e-08 -2.834376e-07 -6.546455e-06
## [56,] 1.773287e-07 1.769495e-07 -2.491334e-08 -1.779394e-07 -4.985554e-06
## [57,] -1.756983e-07 2.381651e-07 -1.051070e-07 5.695182e-06 -8.866833e-06
## [58,] -1.657271e-07 1.501298e-07 -7.113805e-08 3.509972e-06 -5.812330e-06
## [59,] -7.666397e-08 1.031346e-07 -3.468609e-08 -7.035241e-07 -3.224205e-06
## [60,] -3.614804e-08 8.414379e-08 -2.399877e-08 -9.519382e-07 -2.567351e-06
## [61,] -3.438891e-08 3.893198e-08 1.232655e-09 -5.381691e-07 -1.279244e-06
## [62,] -9.055465e-07 -6.087623e-08 1.263225e-07 -4.557152e-06 -5.646722e-06
## [63,] -1.835181e-06 -2.615107e-07 -3.499150e-07 -4.848405e-06 -5.345635e-06
## [64,] -4.835596e-07 -1.002105e-06 1.484781e-07 -2.461889e-06 -4.270122e-06
## [65,] -4.107966e-07 1.344995e-07 -5.750123e-07 -3.900716e-07 3.011435e-07
## [66,] -1.998392e-05 -1.397648e-06 -2.320375e-06 5.701084e-05 8.019525e-05
## [,16] [,17] [,18] [,19] [,20]
## [1,] -5.381599e-06 -1.674575e-05 -5.750375e-07 6.691208e-07 -1.667181e-07
## [2,] 3.676965e-07 -1.778038e-06 -6.388654e-08 1.101250e-07 8.788874e-09
## [3,] -2.667658e-07 -1.544202e-06 2.977858e-08 2.470961e-07 4.976082e-10
## [4,] -4.497462e-07 -1.242393e-06 -4.156419e-08 3.684964e-08 -1.320205e-07
## [5,] 3.740820e-07 -2.595720e-06 -1.103108e-07 3.930172e-08 -7.651909e-08
## [6,] 2.950458e-07 -1.698675e-06 -7.183237e-08 2.480451e-08 -8.415158e-09
## [7,] -5.040960e-07 -8.595517e-07 -1.870980e-08 1.741412e-08 4.808187e-09
## [8,] -6.186057e-07 -1.176875e-06 -5.463448e-08 5.040355e-08 -3.180114e-09
## [9,] -3.995322e-07 -7.766387e-07 -4.328024e-08 4.201761e-08 -4.449448e-09
## [10,] -1.520142e-06 -1.597907e-06 -3.028541e-08 5.072285e-08 1.088968e-08
## [11,] -1.686009e-06 -1.716648e-06 -7.172254e-09 7.060838e-08 9.369027e-09
## [12,] -1.190856e-06 -1.243791e-06 -8.914043e-08 2.431847e-08 1.452754e-08
## [13,] -6.151977e-08 1.264394e-07 4.791025e-08 6.511705e-09 -7.452063e-10
## [14,] -3.379213e-05 -8.805917e-07 -1.161310e-06 -1.492399e-06 -9.849688e-07
## [15,] -6.456675e-06 -5.313877e-05 -3.585316e-07 2.140465e-07 5.645669e-10
## [16,] 2.688876e-04 1.189826e-05 -4.273354e-06 -4.495014e-06 -2.339052e-06
## [17,] 1.189826e-05 1.406581e-04 3.308027e-06 -7.500736e-07 -9.309850e-08
## [18,] -4.273354e-06 3.308027e-06 3.992500e-04 -3.970645e-07 -1.517052e-08
## [19,] -4.495014e-06 -7.500736e-07 -3.970645e-07 3.989165e-04 -2.853052e-07
## [20,] -2.339052e-06 -9.309850e-08 -1.517052e-08 -2.853052e-07 3.995278e-04
## [21,] -4.473218e-06 2.287149e-06 1.573681e-07 4.320544e-08 -4.162721e-07
## [22,] -3.196786e-06 9.363954e-07 8.620586e-08 6.333207e-08 1.067930e-08
## [23,] -1.959981e-06 -5.974657e-07 2.814093e-08 3.772485e-08 2.387650e-08
## [24,] -1.289203e-06 -1.774781e-07 5.388351e-08 -7.491040e-09 2.454314e-08
## [25,] -5.289950e-07 -5.101447e-08 2.966041e-08 -1.084643e-08 1.203962e-08
## [26,] -2.971668e-06 -2.719506e-06 -1.354465e-08 6.523846e-08 3.105292e-08
## [27,] -2.551408e-06 -3.080097e-06 9.329887e-08 7.159134e-08 2.495522e-08
## [28,] -2.919469e-06 -7.077973e-07 -2.491005e-07 2.358181e-09 3.884556e-08
## [29,] 5.771028e-07 -4.019215e-07 1.564363e-07 1.083594e-08 -7.434538e-09
## [30,] 5.807605e-06 -1.158864e-05 3.766871e-07 1.834637e-07 -4.703061e-08
## [31,] -1.547432e-06 -6.916594e-06 -1.887727e-08 3.016250e-08 -5.025529e-08
## [32,] -7.232619e-07 -3.181263e-06 -1.041035e-07 -5.148240e-08 -1.052328e-07
## [33,] 4.482534e-06 -6.556305e-06 -2.197204e-07 -1.092377e-07 5.680920e-09
## [34,] 2.445647e-06 -3.870918e-06 -1.209667e-07 -7.157145e-08 -1.907456e-08
## [35,] -1.174057e-06 -1.618519e-06 2.808723e-09 -6.496562e-09 1.997100e-08
## [36,] -1.347999e-06 -1.389200e-06 -4.544329e-09 1.425671e-08 1.657466e-08
## [37,] -8.008813e-07 -8.779780e-07 -1.090258e-08 1.753003e-08 7.469433e-09
## [38,] -5.712441e-06 -5.476238e-06 -3.321526e-08 1.166769e-07 5.848033e-08
## [39,] -6.248217e-06 -4.990945e-06 1.623669e-08 1.208736e-07 6.105675e-08
## [40,] -2.685375e-06 -4.529248e-06 -1.102905e-07 4.855430e-08 4.135475e-08
## [41,] -4.858983e-07 1.067289e-06 7.645460e-08 -2.906262e-09 1.784942e-09
## [42,] -2.538317e-05 2.662167e-05 -2.794567e-06 -1.363861e-06 4.346824e-08
## [43,] -1.991620e-05 -1.895548e-06 -1.454015e-06 -3.392569e-06 -7.543024e-07
## [44,] -8.154510e-06 -3.414253e-07 -6.292772e-08 -7.977661e-07 -1.110705e-06
## [45,] -1.634660e-05 9.752626e-06 5.716008e-07 2.223998e-07 -1.139175e-06
## [46,] -1.021998e-05 4.035744e-06 2.900836e-07 1.983088e-07 3.956704e-08
## [47,] -4.772521e-06 -1.614030e-06 7.873627e-08 6.409914e-08 6.053195e-08
## [48,] -3.287756e-06 -1.562192e-06 6.295119e-08 2.210313e-08 4.763843e-08
## [49,] -1.770460e-06 -1.095463e-06 2.352955e-08 1.594709e-08 2.405802e-08
## [50,] -1.183009e-05 -1.125426e-05 -6.451247e-08 2.442020e-07 1.214519e-07
## [51,] -1.030496e-05 -1.357657e-05 2.142936e-07 2.802819e-07 1.051331e-07
## [52,] -1.119531e-05 -9.018254e-07 -6.593573e-07 1.162043e-08 1.289639e-07
## [53,] 2.762244e-06 -2.495408e-06 4.053066e-07 4.095445e-08 -2.842925e-08
## [54,] 5.044793e-05 -8.426471e-05 4.826079e-06 1.025338e-06 -5.466062e-07
## [55,] -6.278792e-06 -3.788615e-05 1.823378e-07 -1.514062e-06 -6.219568e-07
## [56,] -2.674165e-06 -1.194133e-05 -4.987145e-07 -6.541308e-07 -5.997125e-08
## [57,] 2.142850e-05 -2.470521e-05 -8.684295e-07 -5.301257e-07 4.309992e-07
## [58,] 1.051319e-05 -1.359908e-05 -4.247013e-07 -3.245104e-07 -5.912401e-08
## [59,] -3.900965e-06 -4.631592e-06 2.689912e-08 -1.628804e-08 6.387016e-08
## [60,] -4.281018e-06 -3.365966e-06 4.265312e-08 2.008900e-08 6.043807e-08
## [61,] -2.816439e-06 -2.423329e-06 7.060323e-09 3.427589e-08 3.434482e-08
## [62,] -2.404866e-05 -2.317753e-05 -1.407543e-07 4.934026e-07 2.454825e-07
## [63,] -2.820111e-05 -1.903181e-05 -2.593597e-07 4.421804e-07 2.795615e-07
## [64,] -4.957648e-06 -2.326876e-05 3.029971e-07 3.388137e-07 9.307792e-08
## [65,] -4.255576e-06 7.821308e-06 -1.505735e-07 -1.051859e-07 3.258090e-08
## [66,] 9.052213e-05 1.570192e-04 1.017970e-05 -1.238721e-05 1.297193e-06
## [,21] [,22] [,23] [,24] [,25]
## [1,] -1.576359e-07 2.135272e-07 -1.467427e-07 1.796292e-07 2.113376e-07
## [2,] -4.657183e-08 -2.217752e-08 6.153365e-09 -8.322053e-09 -6.420554e-09
## [3,] -4.760074e-08 -1.357117e-08 7.647601e-09 -1.707008e-08 -1.238423e-08
## [4,] -1.065255e-07 -7.481108e-09 5.037906e-09 -4.730141e-08 -3.168123e-08
## [5,] 6.655278e-08 1.454594e-07 1.330210e-08 -8.929771e-08 -5.813182e-08
## [6,] 1.106016e-07 1.734109e-07 -3.273093e-08 -7.154553e-08 -3.400915e-08
## [7,] -9.773000e-09 -4.358655e-08 -1.737268e-07 -5.151375e-08 7.635810e-11
## [8,] -4.770929e-08 -2.322016e-08 -4.131356e-08 3.156510e-07 1.599008e-07
## [9,] -3.686993e-08 -2.404004e-10 5.624809e-09 1.411342e-07 1.585759e-07
## [10,] -1.901321e-08 6.596301e-09 1.587341e-08 -2.242782e-08 2.066964e-08
## [11,] -2.483240e-08 1.242561e-08 1.404120e-08 -5.485238e-08 -3.564940e-08
## [12,] -3.007519e-09 3.620781e-09 1.926995e-08 9.563913e-09 3.818931e-09
## [13,] 1.541119e-09 3.570675e-09 -1.462228e-09 -8.460010e-09 -5.153329e-09
## [14,] -1.760698e-06 -1.607377e-06 -1.256562e-06 -1.253758e-06 -5.729760e-07
## [15,] 6.145698e-07 4.636755e-07 -2.052442e-07 3.236918e-07 2.776068e-07
## [16,] -4.473218e-06 -3.196786e-06 -1.959981e-06 -1.289203e-06 -5.289950e-07
## [17,] 2.287149e-06 9.363954e-07 -5.974657e-07 -1.774781e-07 -5.101447e-08
## [18,] 1.573681e-07 8.620586e-08 2.814093e-08 5.388351e-08 2.966041e-08
## [19,] 4.320544e-08 6.333207e-08 3.772485e-08 -7.491040e-09 -1.084643e-08
## [20,] -4.162721e-07 1.067930e-08 2.387650e-08 2.454314e-08 1.203962e-08
## [21,] 3.986986e-04 -4.659723e-07 -1.161266e-08 5.186349e-08 2.918742e-08
## [22,] -4.659723e-07 3.991579e-04 -5.186432e-07 -5.174446e-08 6.888680e-09
## [23,] -1.161266e-08 -5.186432e-07 3.992758e-04 -2.300199e-07 -8.719927e-09
## [24,] 5.186349e-08 -5.174446e-08 -2.300199e-07 3.990243e-04 -3.079110e-07
## [25,] 2.918742e-08 6.888680e-09 -8.719927e-09 -3.079110e-07 3.997424e-04
## [26,] 6.356204e-09 1.831286e-08 3.929707e-08 1.677693e-08 -1.419907e-07
## [27,] -1.385293e-08 8.338002e-09 3.492448e-08 1.167131e-08 -1.481140e-08
## [28,] 6.551929e-08 3.040045e-08 4.014348e-08 8.438347e-08 4.340656e-08
## [29,] -2.400916e-08 -1.106652e-08 -5.215098e-09 -1.636788e-08 -5.743865e-09
## [30,] -3.283011e-07 -1.789873e-07 6.767352e-09 -8.889861e-09 -4.837008e-09
## [31,] -1.202654e-07 -4.875052e-08 4.577485e-08 3.250078e-08 1.475350e-08
## [32,] -3.686544e-08 -3.984315e-08 2.151647e-08 5.187452e-09 1.058034e-10
## [33,] 5.846461e-07 4.600201e-07 3.645556e-08 6.774709e-08 4.341578e-08
## [34,] 5.066023e-07 4.563735e-07 -1.120902e-07 2.260961e-08 3.881797e-08
## [35,] 1.881500e-08 -1.459194e-07 -3.562454e-07 -3.986560e-08 3.680993e-08
## [36,] -2.214930e-09 -4.216020e-08 -8.889702e-08 -1.446716e-07 3.849369e-08
## [37,] -6.262487e-09 1.411177e-09 6.122364e-09 2.366155e-08 4.911803e-08
## [38,] 3.475013e-09 2.652187e-08 7.552528e-08 5.669662e-08 -9.729560e-08
## [39,] 1.954770e-08 4.015152e-08 7.527859e-08 4.088950e-08 -2.888518e-09
## [40,] -8.722976e-09 -1.153581e-08 5.865312e-08 1.105005e-07 5.820959e-08
## [41,] 2.550457e-08 1.788072e-08 -2.904708e-09 -1.352358e-08 -3.524430e-09
## [42,] 1.023396e-06 5.602787e-07 1.128830e-07 2.286727e-07 1.246745e-07
## [43,] 2.591966e-07 2.761953e-07 1.571783e-07 -3.064501e-08 -4.517390e-08
## [44,] -1.138540e-06 5.636505e-08 7.768005e-08 7.240014e-08 3.455634e-08
## [45,] -3.808441e-06 -1.281385e-06 2.902285e-09 1.035367e-07 5.277186e-08
## [46,] -1.351679e-06 -1.952804e-06 -1.091196e-06 -1.025890e-07 1.007288e-08
## [47,] -1.586854e-08 -1.114577e-06 -1.411549e-06 -2.544525e-07 2.519372e-08
## [48,] 6.488392e-08 -1.073692e-07 -3.059675e-07 -1.451946e-06 -4.157482e-07
## [49,] 2.983599e-08 1.423586e-08 -4.391350e-09 -4.284246e-07 -2.531741e-07
## [50,] 1.314663e-08 5.931862e-08 1.557805e-07 1.079229e-07 -2.690149e-07
## [51,] -6.675680e-08 1.859184e-08 1.496162e-07 8.613689e-08 -6.942056e-09
## [52,] 2.463119e-07 1.313031e-07 1.250878e-07 2.255376e-07 1.152698e-07
## [53,] -1.092586e-07 -5.740604e-08 -1.603083e-08 -3.921166e-08 -1.161996e-08
## [54,] -2.596819e-06 -1.381563e-06 -6.245206e-08 -3.324395e-07 -1.895262e-07
## [55,] -6.547574e-07 -2.594616e-07 2.164174e-07 5.192587e-08 5.233199e-09
## [56,] 3.332958e-07 -1.207692e-07 7.876975e-08 3.409447e-08 1.057469e-08
## [57,] 3.223392e-06 1.863611e-06 8.071581e-08 3.376918e-07 2.228410e-07
## [58,] 2.162261e-06 1.537983e-06 -4.691899e-07 1.719166e-07 1.839774e-07
## [59,] 8.548765e-08 -5.458938e-07 -1.079924e-06 -9.736516e-08 1.036046e-07
## [60,] 4.714146e-08 -1.084068e-07 -2.511644e-07 -1.161458e-06 -2.144264e-07
## [61,] 1.956972e-08 1.261988e-08 1.396718e-08 -2.387686e-07 -1.274245e-07
## [62,] 1.759972e-08 1.144554e-07 3.163913e-07 2.278124e-07 -4.364556e-07
## [63,] 1.741887e-07 2.064756e-07 3.302886e-07 2.360548e-07 3.536161e-08
## [64,] -2.830533e-07 -1.538591e-07 1.830531e-07 2.777176e-07 1.376160e-07
## [65,] 2.265740e-07 1.261473e-07 -1.008803e-09 3.991976e-09 2.015816e-08
## [66,] 7.913723e-06 -1.554536e-06 9.969276e-07 1.880251e-05 1.202901e-05
## [,26] [,27] [,28] [,29] [,30]
## [1,] 2.305971e-07 1.416995e-07 -8.428031e-07 6.629177e-08 -2.068694e-08
## [2,] 3.659493e-08 7.252342e-08 -6.210066e-08 4.842842e-08 -6.364691e-07
## [3,] 3.794353e-08 4.696737e-08 -1.041957e-09 1.018248e-08 -1.689326e-07
## [4,] 3.333505e-08 4.327278e-08 -2.613757e-08 1.242769e-08 7.498691e-08
## [5,] 5.261776e-08 7.285785e-08 -5.768223e-08 2.509570e-08 2.557167e-07
## [6,] 3.557977e-08 4.845908e-08 -3.076281e-08 1.558277e-08 1.755555e-07
## [7,] 2.886154e-08 3.255468e-08 7.841754e-09 3.937542e-09 8.259817e-08
## [8,] 4.143373e-08 4.749011e-08 -4.677624e-08 1.675159e-08 1.043146e-07
## [9,] 6.363559e-08 3.421820e-08 -4.647118e-08 1.419238e-08 6.302210e-08
## [10,] -5.299725e-08 -6.824987e-08 -4.843687e-08 1.731877e-08 1.144692e-07
## [11,] -7.381944e-08 -2.595183e-07 -8.238001e-08 -3.672564e-08 2.172762e-07
## [12,] -1.410182e-09 -9.924655e-09 -2.452305e-07 1.560409e-09 -1.628832e-07
## [13,] 6.167946e-09 -5.159390e-08 -8.279749e-09 -2.167409e-07 1.373135e-07
## [14,] -9.838415e-07 -6.017925e-07 -1.196349e-06 2.077234e-07 1.847011e-07
## [15,] -6.433824e-07 -6.803010e-07 -7.070486e-07 -5.367791e-08 -8.150956e-07
## [16,] -2.971668e-06 -2.551408e-06 -2.919469e-06 5.771028e-07 5.807605e-06
## [17,] -2.719506e-06 -3.080097e-06 -7.077973e-07 -4.019215e-07 -1.158864e-05
## [18,] -1.354465e-08 9.329887e-08 -2.491005e-07 1.564363e-07 3.766871e-07
## [19,] 6.523846e-08 7.159134e-08 2.358181e-09 1.083594e-08 1.834637e-07
## [20,] 3.105292e-08 2.495522e-08 3.884556e-08 -7.434538e-09 -4.703061e-08
## [21,] 6.356204e-09 -1.385293e-08 6.551929e-08 -2.400916e-08 -3.283011e-07
## [22,] 1.831286e-08 8.338002e-09 3.040045e-08 -1.106652e-08 -1.789873e-07
## [23,] 3.929707e-08 3.492448e-08 4.014348e-08 -5.215098e-09 6.767352e-09
## [24,] 1.677693e-08 1.167131e-08 8.438347e-08 -1.636788e-08 -8.889861e-09
## [25,] -1.419907e-07 -1.481140e-08 4.340656e-08 -5.743865e-09 -4.837008e-09
## [26,] 3.993382e-04 -3.243390e-07 -8.871558e-08 4.690930e-08 1.772339e-07
## [27,] -3.243390e-07 3.993816e-04 -7.429710e-08 -1.265836e-07 2.348146e-07
## [28,] -8.871558e-08 -7.429710e-08 3.995015e-04 9.082251e-08 -2.016037e-08
## [29,] 4.690930e-08 -1.265836e-07 9.082251e-08 3.996654e-04 6.976405e-08
## [30,] 1.772339e-07 2.348146e-07 -2.016037e-08 6.976405e-08 3.976542e-04
## [31,] 1.672299e-07 1.756201e-07 1.015284e-07 5.087631e-09 -5.532627e-07
## [32,] 8.101355e-08 9.137120e-08 2.964819e-08 1.149149e-08 2.006360e-07
## [33,] 1.007769e-07 1.198486e-07 7.565515e-08 1.825785e-08 7.849665e-07
## [34,] 6.511219e-08 7.420824e-08 6.776749e-08 7.180697e-09 4.711641e-07
## [35,] 5.813683e-08 5.361554e-08 8.830609e-08 -8.040387e-09 1.585782e-07
## [36,] 5.741802e-08 5.203898e-08 5.370290e-08 -2.412603e-09 1.243522e-07
## [37,] -9.029015e-08 1.307418e-08 1.056043e-08 6.770048e-09 7.097700e-08
## [38,] -8.563156e-07 -4.297041e-07 -9.724351e-08 6.825754e-08 3.758501e-07
## [39,] -4.398771e-07 -9.275960e-07 -2.362262e-07 -1.602789e-07 6.293532e-07
## [40,] -7.981765e-08 -2.112354e-07 -4.843871e-07 1.729481e-08 -3.858961e-07
## [41,] 4.647599e-08 -1.846693e-07 8.153674e-09 -3.504054e-07 3.380826e-07
## [42,] -2.419825e-07 1.546752e-08 -7.050031e-07 3.964867e-07 4.582853e-06
## [43,] 2.610369e-07 2.995736e-07 -1.714716e-08 6.123378e-08 1.269205e-06
## [44,] 1.035941e-07 8.529314e-08 1.191817e-07 -2.235366e-08 -2.248376e-07
## [45,] -1.709227e-08 -8.346404e-08 1.508528e-07 -7.717737e-08 -1.350155e-06
## [46,] 3.146812e-08 8.267130e-11 7.240050e-08 -3.509818e-08 -6.661479e-07
## [47,] 9.848971e-08 8.366984e-08 1.359680e-07 -1.945446e-08 4.443899e-08
## [48,] 8.034363e-08 6.640226e-08 1.411659e-07 -2.061800e-08 8.423570e-08
## [49,] -2.727859e-07 -3.694069e-09 5.755302e-08 2.476530e-09 6.611741e-08
## [50,] -1.944505e-06 -9.648042e-07 -2.224876e-07 1.600433e-07 7.577968e-07
## [51,] -9.430926e-07 -1.840999e-06 -3.925516e-07 -3.272395e-07 6.855576e-07
## [52,] -3.023790e-07 -4.929527e-07 -1.086817e-06 1.901244e-07 6.322135e-07
## [53,] 1.665966e-07 -3.217512e-07 1.955203e-07 -5.903691e-07 -1.602470e-07
## [54,] 1.151382e-06 1.101964e-06 7.521414e-07 -1.573108e-07 -1.500402e-05
## [55,] 8.686536e-07 8.641482e-07 5.590533e-07 -3.994475e-08 -3.334986e-06
## [56,] 2.971640e-07 3.338230e-07 1.151858e-07 4.061419e-08 7.298997e-07
## [57,] 3.226996e-07 3.894950e-07 3.152214e-07 6.082253e-08 3.066483e-06
## [58,] 2.005198e-07 2.249367e-07 2.740641e-07 1.532146e-08 1.685436e-06
## [59,] 1.672678e-07 1.500677e-07 2.678429e-07 -2.796424e-08 4.218915e-07
## [60,] 1.460785e-07 1.257839e-07 2.056810e-07 -2.247028e-08 2.707661e-07
## [61,] -4.547424e-07 6.243348e-09 7.689765e-08 1.337061e-08 1.797335e-07
## [62,] -3.639177e-06 -1.855682e-06 -4.105976e-07 3.241082e-07 1.572402e-06
## [63,] -1.908531e-06 -3.519433e-06 -1.079069e-06 -4.183381e-07 2.802624e-06
## [64,] -3.526316e-07 -1.014600e-06 -1.219440e-06 1.316605e-08 -2.063006e-06
## [65,] 1.772738e-07 -5.789598e-07 -3.266018e-08 -7.313694e-07 1.615590e-06
## [66,] -3.574090e-06 -6.212121e-06 1.501326e-05 -3.646391e-06 -8.109113e-06
## [,31] [,32] [,33] [,34] [,35]
## [1,] -6.934362e-07 -6.556152e-07 -1.680073e-06 -1.648618e-06 -1.489113e-06
## [2,] -2.509855e-07 1.371652e-08 1.197399e-07 7.303616e-08 2.992818e-08
## [3,] -9.715332e-07 -2.455642e-07 7.260871e-08 5.210023e-08 2.126823e-08
## [4,] -2.225146e-07 -4.316792e-07 -4.056919e-07 -1.049083e-08 -1.251886e-08
## [5,] 1.284143e-07 -3.438460e-07 -1.265041e-06 -4.872420e-07 -5.861654e-08
## [6,] 9.985607e-08 4.576997e-08 -4.400859e-07 -8.387073e-07 -5.307712e-07
## [7,] 6.482915e-08 3.702257e-08 2.057449e-08 -4.895062e-07 -7.173043e-07
## [8,] 7.050605e-08 4.912109e-08 -8.131006e-09 -8.284154e-08 -2.813866e-07
## [9,] 3.883455e-08 3.105309e-08 -3.027241e-08 -3.213951e-08 -5.765654e-08
## [10,] 9.726996e-08 5.466462e-08 3.106650e-08 1.348251e-08 6.769729e-09
## [11,] 9.170936e-08 5.261459e-08 -1.077104e-08 -2.093552e-08 -2.034800e-08
## [12,] 8.584167e-08 5.004448e-08 7.234099e-08 4.963683e-08 3.901713e-08
## [13,] -8.496671e-09 -6.533962e-09 -2.455746e-08 -1.812585e-08 -1.212006e-08
## [14,] 3.319922e-07 -1.234361e-07 1.134604e-06 9.067545e-07 -2.170200e-07
## [15,] -2.023334e-06 -1.640479e-06 -2.840721e-06 -2.254085e-06 -1.525789e-06
## [16,] -1.547432e-06 -7.232619e-07 4.482534e-06 2.445647e-06 -1.174057e-06
## [17,] -6.916594e-06 -3.181263e-06 -6.556305e-06 -3.870918e-06 -1.618519e-06
## [18,] -1.887727e-08 -1.041035e-07 -2.197204e-07 -1.209667e-07 2.808723e-09
## [19,] 3.016250e-08 -5.148240e-08 -1.092377e-07 -7.157145e-08 -6.496562e-09
## [20,] -5.025529e-08 -1.052328e-07 5.680920e-09 -1.907456e-08 1.997100e-08
## [21,] -1.202654e-07 -3.686544e-08 5.846461e-07 5.066023e-07 1.881500e-08
## [22,] -4.875052e-08 -3.984315e-08 4.600201e-07 4.563735e-07 -1.459194e-07
## [23,] 4.577485e-08 2.151647e-08 3.645556e-08 -1.120902e-07 -3.562454e-07
## [24,] 3.250078e-08 5.187452e-09 6.774709e-08 2.260961e-08 -3.986560e-08
## [25,] 1.475350e-08 1.058034e-10 4.341578e-08 3.881797e-08 3.680993e-08
## [26,] 1.672299e-07 8.101355e-08 1.007769e-07 6.511219e-08 5.813683e-08
## [27,] 1.756201e-07 9.137120e-08 1.198486e-07 7.420824e-08 5.361554e-08
## [28,] 1.015284e-07 2.964819e-08 7.565515e-08 6.776749e-08 8.830609e-08
## [29,] 5.087631e-09 1.149149e-08 1.825785e-08 7.180697e-09 -8.040387e-09
## [30,] -5.532627e-07 2.006360e-07 7.849665e-07 4.711641e-07 1.585782e-07
## [31,] 3.970439e-04 -6.274196e-07 4.072655e-07 2.653300e-07 1.398439e-07
## [32,] -6.274196e-07 3.989709e-04 -9.872829e-07 9.493283e-08 6.484070e-08
## [33,] 4.072655e-07 -9.872829e-07 3.966238e-04 -1.094630e-06 1.252737e-07
## [34,] 2.653300e-07 9.493283e-08 -1.094630e-06 3.982033e-04 -9.740398e-07
## [35,] 1.398439e-07 6.484070e-08 1.252737e-07 -9.740398e-07 3.986554e-04
## [36,] 1.148684e-07 5.541028e-08 1.151254e-07 -4.380649e-08 -2.886213e-07
## [37,] 6.177324e-08 3.159441e-08 4.437430e-08 3.085489e-08 -1.026476e-08
## [38,] 3.467484e-07 1.674633e-07 2.329397e-07 1.535628e-07 1.281750e-07
## [39,] 3.299752e-07 1.473733e-07 1.710518e-07 1.116587e-07 1.073141e-07
## [40,] 2.701866e-07 1.530991e-07 3.414513e-07 2.381502e-07 1.728864e-07
## [41,] -3.979234e-08 -3.606554e-08 -9.245353e-08 -6.049719e-08 -2.913861e-08
## [42,] -7.093730e-07 -9.417251e-07 -1.769878e-06 -1.020197e-06 -1.538001e-07
## [43,] -1.576309e-06 -6.201270e-07 -4.580305e-07 -3.202783e-07 -2.376187e-08
## [44,] -6.143412e-07 -7.227421e-08 4.341632e-07 -6.771821e-08 5.608079e-08
## [45,] -5.364341e-07 2.347454e-07 3.007795e-06 1.954273e-06 -6.316658e-08
## [46,] -2.174373e-07 -1.525624e-07 1.949270e-06 1.544081e-06 -6.048024e-07
## [47,] 1.362750e-07 5.696726e-08 1.907992e-07 -3.972538e-07 -1.050251e-06
## [48,] 1.363710e-07 5.400470e-08 1.515102e-07 1.082126e-08 -1.942779e-07
## [49,] 8.386901e-08 3.472416e-08 8.397713e-08 6.710632e-08 3.596127e-08
## [50,] 7.015008e-07 3.373029e-07 4.610500e-07 3.024373e-07 2.562936e-07
## [51,] 7.562487e-07 4.055767e-07 5.983249e-07 3.800437e-07 2.676613e-07
## [52,] 3.215301e-07 3.998505e-08 6.976673e-08 1.003800e-07 2.229780e-07
## [53,] 3.162457e-08 7.584082e-08 1.577699e-07 8.609202e-08 2.555938e-10
## [54,] -1.211704e-06 1.978843e-06 5.435712e-06 3.189851e-06 8.551447e-07
## [55,] -1.108275e-05 -1.970608e-06 2.078081e-06 1.261379e-06 5.991060e-07
## [56,] -2.367687e-06 -3.129932e-06 -3.194895e-06 3.502535e-07 2.421632e-07
## [57,] 1.559633e-06 -3.155346e-06 -1.199785e-05 -3.935477e-06 6.372363e-07
## [58,] 9.160042e-07 3.299682e-07 -4.120006e-06 -5.316665e-06 -2.251115e-06
## [59,] 3.868689e-07 1.737771e-07 3.340118e-07 -2.427022e-06 -3.276068e-06
## [60,] 2.873282e-07 1.256253e-07 3.290249e-07 -6.664226e-08 -5.379706e-07
## [61,] 1.792454e-07 8.187929e-08 1.659562e-07 1.246521e-07 3.320255e-08
## [62,] 1.446178e-06 6.951382e-07 9.622635e-07 6.313910e-07 5.283783e-07
## [63,] 1.372505e-06 5.562663e-07 6.438341e-07 4.402311e-07 4.795833e-07
## [64,] 1.076587e-06 7.442066e-07 1.534439e-06 1.004665e-06 5.894727e-07
## [65,] -2.148892e-07 -2.482360e-07 -5.315536e-07 -3.223687e-07 -1.062814e-07
## [66,] -3.141515e-06 -4.360917e-06 1.755875e-05 1.581941e-05 1.449813e-05
## [,36] [,37] [,38] [,39] [,40]
## [1,] -1.485716e-06 -6.111615e-07 -4.960276e-07 -6.098736e-08 -1.422680e-06
## [2,] 2.933567e-08 1.764508e-08 7.918334e-08 1.847208e-07 -2.115330e-07
## [3,] 2.715551e-08 1.868058e-08 8.092667e-08 7.662625e-08 4.941677e-08
## [4,] 1.060371e-08 1.694841e-08 6.500020e-08 6.637148e-08 5.765748e-09
## [5,] 1.463767e-08 2.824581e-08 1.036678e-07 1.045900e-07 4.617188e-09
## [6,] -4.936159e-08 2.035787e-08 7.239620e-08 7.057854e-08 1.612564e-08
## [7,] -2.293525e-07 -6.138817e-09 6.055110e-08 5.704087e-08 3.951713e-08
## [8,] -1.044730e-06 -3.356500e-07 5.929561e-08 7.272529e-08 -2.460434e-08
## [9,] -3.582164e-07 -2.737345e-07 -1.117459e-07 2.984311e-08 -4.478629e-08
## [10,] 1.730598e-08 -1.291919e-07 -6.463108e-07 -3.100082e-07 -1.050419e-07
## [11,] 1.180964e-08 4.711290e-09 -3.130595e-07 -6.022644e-07 -1.179901e-07
## [12,] 3.563863e-08 1.416982e-08 -4.771561e-08 -3.206388e-08 -5.098472e-07
## [13,] -6.958137e-09 2.325194e-09 3.825772e-08 -1.325704e-07 8.263262e-08
## [14,] -2.394025e-08 2.816994e-08 -1.063770e-06 -1.152608e-06 -9.644929e-07
## [15,] -1.467711e-06 -6.802049e-07 -1.577589e-06 -1.307899e-06 -1.631372e-06
## [16,] -1.347999e-06 -8.008813e-07 -5.712441e-06 -6.248217e-06 -2.685375e-06
## [17,] -1.389200e-06 -8.779780e-07 -5.476238e-06 -4.990945e-06 -4.529248e-06
## [18,] -4.544329e-09 -1.090258e-08 -3.321526e-08 1.623669e-08 -1.102905e-07
## [19,] 1.425671e-08 1.753003e-08 1.166769e-07 1.208736e-07 4.855430e-08
## [20,] 1.657466e-08 7.469433e-09 5.848033e-08 6.105675e-08 4.135475e-08
## [21,] -2.214930e-09 -6.262487e-09 3.475013e-09 1.954770e-08 -8.722976e-09
## [22,] -4.216020e-08 1.411177e-09 2.652187e-08 4.015152e-08 -1.153581e-08
## [23,] -8.889702e-08 6.122364e-09 7.552528e-08 7.527859e-08 5.865312e-08
## [24,] -1.446716e-07 2.366155e-08 5.669662e-08 4.088950e-08 1.105005e-07
## [25,] 3.849369e-08 4.911803e-08 -9.729560e-08 -2.888518e-09 5.820959e-08
## [26,] 5.741802e-08 -9.029015e-08 -8.563156e-07 -4.398771e-07 -7.981765e-08
## [27,] 5.203898e-08 1.307418e-08 -4.297041e-07 -9.275960e-07 -2.112354e-07
## [28,] 5.370290e-08 1.056043e-08 -9.724351e-08 -2.362262e-07 -4.843871e-07
## [29,] -2.412603e-09 6.770048e-09 6.825754e-08 -1.602789e-07 1.729481e-08
## [30,] 1.243522e-07 7.097700e-08 3.758501e-07 6.293532e-07 -3.858961e-07
## [31,] 1.148684e-07 6.177324e-08 3.467484e-07 3.299752e-07 2.701866e-07
## [32,] 5.541028e-08 3.159441e-08 1.674633e-07 1.473733e-07 1.530991e-07
## [33,] 1.151254e-07 4.437430e-08 2.329397e-07 1.710518e-07 3.414513e-07
## [34,] -4.380649e-08 3.085489e-08 1.535628e-07 1.116587e-07 2.381502e-07
## [35,] -2.886213e-07 -1.026476e-08 1.281750e-07 1.073141e-07 1.728864e-07
## [36,] 3.984856e-04 -4.615721e-07 1.120747e-07 1.005145e-07 1.165382e-07
## [37,] -4.615721e-07 3.997274e-04 -2.555887e-07 1.575510e-08 3.465388e-08
## [38,] 1.120747e-07 -2.555887e-07 3.980490e-04 -9.957723e-07 -1.781969e-07
## [39,] 1.005145e-07 1.575510e-08 -9.957723e-07 3.980844e-04 -3.817525e-07
## [40,] 1.165382e-07 3.465388e-08 -1.781969e-07 -3.817525e-07 3.989885e-04
## [41,] -1.844914e-08 3.941138e-09 1.124840e-07 -3.711970e-07 1.588797e-07
## [42,] -1.433155e-07 -1.162373e-07 -5.270361e-07 -6.223984e-07 3.199326e-08
## [43,] 6.424889e-08 7.359097e-08 4.711540e-07 4.287430e-07 3.401782e-07
## [44,] 5.084019e-08 2.482573e-08 1.945076e-07 2.030156e-07 1.316978e-07
## [45,] -6.092605e-08 -3.298132e-08 -7.614165e-08 4.801616e-09 -1.953315e-07
## [46,] -1.436409e-07 -2.746525e-09 3.170373e-08 8.195203e-08 -9.749151e-08
## [47,] -2.512491e-07 -3.014952e-09 1.973376e-07 1.865109e-07 2.039567e-07
## [48,] -1.243999e-06 -2.845004e-07 1.751967e-07 1.499560e-07 2.207371e-07
## [49,] -2.656853e-07 -1.510042e-07 -4.435139e-07 1.157722e-08 9.688268e-08
## [50,] 2.319485e-07 -4.118895e-07 -3.664743e-06 -1.929948e-06 -3.286002e-07
## [51,] 2.382349e-07 6.442480e-08 -1.837701e-06 -3.495743e-06 -9.532017e-07
## [52,] 1.427327e-07 2.573549e-08 -5.782582e-07 -1.266774e-06 -1.310331e-06
## [53,] 6.715513e-09 2.994976e-08 3.386214e-07 -4.090963e-07 3.495864e-08
## [54,] 7.230974e-07 4.697526e-07 2.425646e-06 3.603348e-06 -1.548005e-06
## [55,] 5.293449e-07 3.143642e-07 1.773053e-06 1.808921e-06 1.009766e-06
## [56,] 2.000907e-07 1.119853e-07 6.127218e-07 5.357647e-07 5.772158e-07
## [57,] 4.392825e-07 1.426035e-07 7.787123e-07 5.183147e-07 1.350634e-06
## [58,] -1.939002e-08 8.983962e-08 4.882983e-07 3.265565e-07 8.865828e-07
## [59,] -5.722497e-07 -1.566573e-08 3.652780e-07 3.066522e-07 5.035681e-07
## [60,] -3.324404e-06 -9.826514e-07 3.055832e-07 2.612635e-07 3.725225e-07
## [61,] -9.545774e-07 -5.713729e-07 -8.970551e-07 1.930788e-08 1.550770e-07
## [62,] 4.708211e-07 -8.099735e-07 -7.532276e-06 -4.122888e-06 -7.109563e-07
## [63,] 4.264268e-07 1.020723e-07 -4.140365e-06 -7.425711e-06 -1.736764e-06
## [64,] 4.236327e-07 1.615463e-07 -7.536374e-07 -1.808145e-06 -2.889926e-06
## [65,] -8.137951e-08 -8.717895e-09 3.991930e-07 -1.171072e-06 3.912976e-07
## [66,] 4.450066e-06 -2.015044e-06 -4.769992e-06 -8.082144e-06 1.737034e-05
## [,41] [,42] [,43] [,44] [,45]
## [1,] 2.122005e-07 -3.385838e-06 1.596640e-06 -6.452458e-08 -3.836549e-08
## [2,] 1.470532e-07 2.224558e-07 2.573600e-07 -3.003364e-09 -1.882791e-07
## [3,] -9.390409e-09 -3.590062e-07 6.565460e-08 -5.889629e-08 -1.554138e-07
## [4,] -3.158401e-09 -3.513994e-07 8.925257e-08 -1.107412e-07 -9.362926e-08
## [5,] -8.084341e-09 -7.247000e-07 1.813037e-07 -6.305295e-09 4.525258e-07
## [6,] -7.937527e-09 -4.829355e-07 9.467207e-08 -3.096916e-08 3.854712e-07
## [7,] -6.724786e-09 -1.604540e-07 7.353569e-08 1.471957e-08 -6.172124e-08
## [8,] 3.538177e-09 -3.245248e-07 2.116831e-07 -3.656012e-09 -1.075114e-07
## [9,] 1.028622e-08 -2.370370e-07 1.751692e-07 -8.063586e-09 -6.924746e-08
## [10,] 4.032289e-08 -2.383910e-07 2.094606e-07 3.966682e-08 -5.031207e-08
## [11,] -1.281785e-07 -2.169825e-07 2.771580e-07 3.816294e-08 -4.114458e-08
## [12,] 8.838714e-08 -2.793952e-07 1.253527e-07 4.646769e-08 -3.526498e-08
## [13,] -3.317733e-07 1.046587e-07 1.282627e-08 -1.531131e-09 1.801878e-08
## [14,] -9.968195e-08 -3.255198e-06 -4.419128e-06 -2.442973e-06 -4.939579e-06
## [15,] 2.001125e-07 -1.537808e-06 -5.734921e-07 1.359063e-07 2.612450e-06
## [16,] -4.858983e-07 -2.538317e-05 -1.991620e-05 -8.154510e-06 -1.634660e-05
## [17,] 1.067289e-06 2.662167e-05 -1.895548e-06 -3.414253e-07 9.752626e-06
## [18,] 7.645460e-08 -2.794567e-06 -1.454015e-06 -6.292772e-08 5.716008e-07
## [19,] -2.906262e-09 -1.363861e-06 -3.392569e-06 -7.977661e-07 2.223998e-07
## [20,] 1.784942e-09 4.346824e-08 -7.543024e-07 -1.110705e-06 -1.139175e-06
## [21,] 2.550457e-08 1.023396e-06 2.591966e-07 -1.138540e-06 -3.808441e-06
## [22,] 1.788072e-08 5.602787e-07 2.761953e-07 5.636505e-08 -1.281385e-06
## [23,] -2.904708e-09 1.128830e-07 1.571783e-07 7.768005e-08 2.902285e-09
## [24,] -1.352358e-08 2.286727e-07 -3.064501e-08 7.240014e-08 1.035367e-07
## [25,] -3.524430e-09 1.246745e-07 -4.517390e-08 3.455634e-08 5.277186e-08
## [26,] 4.647599e-08 -2.419825e-07 2.610369e-07 1.035941e-07 -1.709227e-08
## [27,] -1.846693e-07 1.546752e-08 2.995736e-07 8.529314e-08 -8.346404e-08
## [28,] 8.153674e-09 -7.050031e-07 -1.714716e-08 1.191817e-07 1.508528e-07
## [29,] -3.504054e-07 3.964867e-07 6.123378e-08 -2.235366e-08 -7.717737e-08
## [30,] 3.380826e-07 4.582853e-06 1.269205e-06 -2.248376e-07 -1.350155e-06
## [31,] -3.979234e-08 -7.093730e-07 -1.576309e-06 -6.143412e-07 -5.364341e-07
## [32,] -3.606554e-08 -9.417251e-07 -6.201270e-07 -7.227421e-08 2.347454e-07
## [33,] -9.245353e-08 -1.769878e-06 -4.580305e-07 4.341632e-07 3.007795e-06
## [34,] -6.049719e-08 -1.020197e-06 -3.202783e-07 -6.771821e-08 1.954273e-06
## [35,] -2.913861e-08 -1.538001e-07 -2.376187e-08 5.608079e-08 -6.316658e-08
## [36,] -1.844914e-08 -1.433155e-07 6.424889e-08 5.084019e-08 -6.092605e-08
## [37,] 3.941138e-09 -1.162373e-07 7.359097e-08 2.482573e-08 -3.298132e-08
## [38,] 1.124840e-07 -5.270361e-07 4.711540e-07 1.945076e-07 -7.614165e-08
## [39,] -3.711970e-07 -6.223984e-07 4.287430e-07 2.030156e-07 4.801616e-09
## [40,] 1.588797e-07 3.199326e-08 3.401782e-07 1.316978e-07 -1.953315e-07
## [41,] 3.994185e-04 -8.508022e-08 -9.344863e-08 4.614208e-09 1.232485e-07
## [42,] -8.508022e-08 3.822952e-04 -7.592981e-06 -7.817364e-08 3.892717e-06
## [43,] -9.344863e-08 -7.592981e-06 3.864744e-04 -2.905182e-06 1.102330e-06
## [44,] 4.614208e-09 -7.817364e-08 -2.905182e-06 3.965618e-04 -3.744683e-06
## [45,] 1.232485e-07 3.892717e-06 1.102330e-06 -3.744683e-06 3.862773e-04
## [46,] 6.866381e-08 1.961646e-06 8.794519e-07 1.456307e-07 -4.966903e-06
## [47,] -1.899651e-08 2.887198e-07 2.724303e-07 1.931622e-07 -8.673818e-08
## [48,] -2.800867e-08 1.733209e-07 9.418422e-08 1.464040e-07 1.055975e-07
## [49,] 4.675223e-09 2.804180e-08 6.551571e-08 7.494927e-08 4.263591e-08
## [50,] 2.275336e-07 -1.056486e-06 9.813770e-07 4.048152e-07 -1.325775e-07
## [51,] -5.260970e-07 2.444701e-07 1.278223e-06 3.585210e-07 -4.317852e-07
## [52,] -1.243038e-08 -3.559390e-06 -3.148425e-07 3.971234e-07 6.924829e-07
## [53,] -7.357028e-07 1.947036e-06 3.816061e-07 -8.377871e-08 -3.969912e-07
## [54,] 1.501211e-06 4.042948e-05 9.833220e-06 -1.927098e-06 -1.029014e-05
## [55,] -4.671475e-08 6.814852e-07 -1.042498e-05 -3.788115e-06 -2.829246e-06
## [56,] -1.380356e-07 -3.924654e-06 -3.970126e-06 4.335252e-07 2.150008e-06
## [57,] -3.781089e-07 -6.988689e-06 -2.288495e-06 2.667117e-06 1.511396e-05
## [58,] -2.287855e-07 -3.636853e-06 -1.444354e-06 -2.027422e-07 8.684625e-06
## [59,] -8.280607e-08 -3.322107e-07 -6.274099e-08 1.835974e-07 -4.114046e-08
## [60,] -5.568852e-08 -9.998814e-08 9.288262e-08 1.832587e-07 -3.928512e-08
## [61,] 1.122687e-08 -1.553834e-07 1.427460e-07 1.083718e-07 -1.967593e-08
## [62,] 5.097315e-07 -2.219269e-06 1.983805e-06 8.186536e-07 -3.068923e-07
## [63,] -1.052652e-06 -4.324336e-06 1.346999e-06 9.167408e-07 3.165582e-07
## [64,] 4.232365e-07 4.208172e-06 2.411112e-06 3.258134e-07 -1.576663e-06
## [65,] -1.455154e-06 -2.775708e-06 -1.030571e-06 8.561463e-08 9.299378e-07
## [66,] -3.047119e-06 5.023227e-05 -4.959046e-05 2.045903e-06 9.203763e-06
## [,46] [,47] [,48] [,49] [,50]
## [1,] 5.649393e-07 -5.661797e-07 -4.665317e-07 -9.887992e-08 -7.369209e-08
## [2,] -8.476549e-08 1.431463e-08 1.326987e-08 1.077033e-08 1.558924e-07
## [3,] -5.263209e-08 1.170743e-08 3.609210e-09 6.296524e-09 1.592102e-07
## [4,] -3.404431e-08 -1.762456e-08 -3.951104e-08 -1.425080e-08 1.286692e-07
## [5,] 5.003071e-07 -1.592168e-08 -8.094951e-08 -3.045701e-08 2.047467e-07
## [6,] 4.610482e-07 -1.400024e-07 -8.002421e-08 -1.440463e-08 1.419161e-07
## [7,] -1.600412e-07 -3.627030e-07 -9.990984e-08 2.731475e-09 1.190931e-07
## [8,] -4.380679e-08 -1.513211e-07 -2.459550e-07 -6.941742e-09 1.341218e-07
## [9,] 7.612283e-09 -5.047498e-08 -5.083184e-08 1.895353e-08 -4.287190e-08
## [10,] 1.454454e-08 1.705049e-08 -1.867701e-09 -1.179737e-07 -8.236324e-07
## [11,] 3.726954e-08 -5.625748e-09 -4.042888e-08 -3.165965e-08 -4.032367e-07
## [12,] -2.142613e-09 4.928918e-08 4.160931e-08 1.876286e-08 -2.049082e-08
## [13,] 1.423179e-08 -9.863621e-09 -1.364946e-08 -1.443367e-09 5.189167e-08
## [14,] -3.718307e-06 -2.141715e-06 -1.685941e-06 -7.354414e-07 -2.509648e-06
## [15,] 1.498786e-06 -5.134441e-07 -5.009010e-07 -2.441658e-07 -2.717195e-06
## [16,] -1.021998e-05 -4.772521e-06 -3.287756e-06 -1.770460e-06 -1.183009e-05
## [17,] 4.035744e-06 -1.614030e-06 -1.562192e-06 -1.095463e-06 -1.125426e-05
## [18,] 2.900836e-07 7.873627e-08 6.295119e-08 2.352955e-08 -6.451247e-08
## [19,] 1.983088e-07 6.409914e-08 2.210313e-08 1.594709e-08 2.442020e-07
## [20,] 3.956704e-08 6.053195e-08 4.763843e-08 2.405802e-08 1.214519e-07
## [21,] -1.351679e-06 -1.586854e-08 6.488392e-08 2.983599e-08 1.314663e-08
## [22,] -1.952804e-06 -1.114577e-06 -1.073692e-07 1.423586e-08 5.931862e-08
## [23,] -1.091196e-06 -1.411549e-06 -3.059675e-07 -4.391350e-09 1.557805e-07
## [24,] -1.025890e-07 -2.544525e-07 -1.451946e-06 -4.284246e-07 1.079229e-07
## [25,] 1.007288e-08 2.519372e-08 -4.157482e-07 -2.531741e-07 -2.690149e-07
## [26,] 3.146812e-08 9.848971e-08 8.034363e-08 -2.727859e-07 -1.944505e-06
## [27,] 8.267130e-11 8.366984e-08 6.640226e-08 -3.694069e-09 -9.648042e-07
## [28,] 7.240050e-08 1.359680e-07 1.411659e-07 5.755302e-08 -2.224876e-07
## [29,] -3.509818e-08 -1.945446e-08 -2.061800e-08 2.476530e-09 1.600433e-07
## [30,] -6.661479e-07 4.443899e-08 8.423570e-08 6.611741e-08 7.577968e-07
## [31,] -2.174373e-07 1.362750e-07 1.363710e-07 8.386901e-08 7.015008e-07
## [32,] -1.525624e-07 5.696726e-08 5.400470e-08 3.472416e-08 3.373029e-07
## [33,] 1.949270e-06 1.907992e-07 1.515102e-07 8.397713e-08 4.610500e-07
## [34,] 1.544081e-06 -3.972538e-07 1.082126e-08 6.710632e-08 3.024373e-07
## [35,] -6.048024e-07 -1.050251e-06 -1.942779e-07 3.596127e-08 2.562936e-07
## [36,] -1.436409e-07 -2.512491e-07 -1.243999e-06 -2.656853e-07 2.319485e-07
## [37,] -2.746525e-09 -3.014952e-09 -2.845004e-07 -1.510042e-07 -4.118895e-07
## [38,] 3.170373e-08 1.973376e-07 1.751967e-07 -4.435139e-07 -3.664743e-06
## [39,] 8.195203e-08 1.865109e-07 1.499560e-07 1.157722e-08 -1.929948e-06
## [40,] -9.749151e-08 2.039567e-07 2.207371e-07 9.688268e-08 -3.286002e-07
## [41,] 6.866381e-08 -1.899651e-08 -2.800867e-08 4.675223e-09 2.275336e-07
## [42,] 1.961646e-06 2.887198e-07 1.733209e-07 2.804180e-08 -1.056486e-06
## [43,] 8.794519e-07 2.724303e-07 9.418422e-08 6.551571e-08 9.813770e-07
## [44,] 1.456307e-07 1.931622e-07 1.464040e-07 7.494927e-08 4.048152e-07
## [45,] -4.966903e-06 -8.673818e-08 1.055975e-07 4.263591e-08 -1.325775e-07
## [46,] 3.940241e-04 -2.834749e-06 -2.875221e-07 2.396629e-08 8.000912e-08
## [47,] -2.834749e-06 3.965417e-04 -5.812871e-07 2.194529e-08 4.054323e-07
## [48,] -2.875221e-07 -5.812871e-07 3.967164e-04 -9.362848e-07 3.524316e-07
## [49,] 2.396629e-08 2.194529e-08 -9.362848e-07 3.994459e-04 -8.771988e-07
## [50,] 8.000912e-08 4.054323e-07 3.524316e-07 -8.771988e-07 3.924436e-04
## [51,] -6.341564e-08 3.859161e-07 3.313809e-07 5.492312e-08 -3.956906e-06
## [52,] 3.656655e-07 3.961479e-07 3.859590e-07 1.530260e-07 -1.235237e-06
## [53,] -1.958629e-07 -4.908911e-08 -4.169099e-08 2.392137e-08 7.373083e-07
## [54,] -5.038421e-06 -8.665338e-08 1.723785e-07 2.698182e-07 4.900000e-06
## [55,] -1.140386e-06 5.759170e-07 5.400821e-07 3.608912e-07 3.602323e-06
## [56,] -4.520036e-07 2.185525e-07 2.135529e-07 1.350443e-07 1.238242e-06
## [57,] 8.285830e-06 6.829239e-07 6.418158e-07 3.451654e-07 1.537019e-06
## [58,] 5.937705e-06 -1.500440e-06 1.371251e-07 2.638460e-07 9.636324e-07
## [59,] -2.130388e-06 -3.470666e-06 -5.903624e-07 9.952836e-08 7.356637e-07
## [60,] -4.151285e-07 -7.197897e-07 -4.239116e-06 -1.053320e-06 6.204834e-07
## [61,] 6.349848e-09 4.030100e-08 -1.060147e-06 -7.063822e-07 -1.655201e-06
## [62,] 1.422012e-07 8.264272e-07 7.241468e-07 -1.607804e-06 -1.515600e-05
## [63,] 4.687400e-07 8.490124e-07 7.041364e-07 1.453275e-07 -8.704487e-06
## [64,] -7.380616e-07 6.158478e-07 6.698950e-07 3.137329e-07 -1.619122e-06
## [65,] 4.661840e-07 -2.451638e-08 -5.607772e-08 2.721442e-08 8.887222e-07
## [66,] -7.463512e-06 1.460566e-05 2.247994e-05 9.804547e-06 -9.934329e-06
## [,51] [,52] [,53] [,54] [,55]
## [1,] 2.071709e-07 -2.038103e-06 3.406853e-07 6.654122e-06 5.877754e-07
## [2,] 2.228181e-07 -7.384519e-08 8.504126e-08 -2.006310e-06 -6.771396e-07
## [3,] 1.949602e-07 -9.526350e-09 4.122186e-08 3.074057e-07 -2.749600e-06
## [4,] 1.626013e-07 -7.110338e-08 4.218652e-08 8.614761e-07 -3.449398e-07
## [5,] 2.738781e-07 -1.640892e-07 8.736454e-08 2.060453e-06 8.684285e-07
## [6,] 1.871972e-07 -9.120650e-08 5.657706e-08 1.382405e-06 5.905910e-07
## [7,] 1.352034e-07 2.135625e-08 1.691866e-08 5.707265e-07 3.371202e-07
## [8,] 1.640622e-07 -1.126768e-07 4.593994e-08 9.098340e-07 4.635837e-07
## [9,] 8.424377e-08 -1.189516e-07 4.080069e-08 6.159386e-07 2.966482e-07
## [10,] -3.738133e-07 -2.045166e-07 9.051308e-08 8.508736e-07 5.511678e-07
## [11,] -8.619167e-07 -3.840850e-07 -1.328912e-07 1.154484e-06 5.755211e-07
## [12,] -1.276665e-07 -5.534860e-07 4.022511e-08 1.665468e-08 4.311271e-07
## [13,] -1.785099e-07 1.192660e-08 -3.523298e-07 2.678160e-07 -3.118368e-08
## [14,] -2.035078e-06 -2.645357e-06 4.758640e-07 3.137169e-06 -2.834376e-07
## [15,] -2.655996e-06 -2.160539e-06 2.334192e-08 -4.029583e-07 -6.546455e-06
## [16,] -1.030496e-05 -1.119531e-05 2.762244e-06 5.044793e-05 -6.278792e-06
## [17,] -1.357657e-05 -9.018254e-07 -2.495408e-06 -8.426471e-05 -3.788615e-05
## [18,] 2.142936e-07 -6.593573e-07 4.053066e-07 4.826079e-06 1.823378e-07
## [19,] 2.802819e-07 1.162043e-08 4.095445e-08 1.025338e-06 -1.514062e-06
## [20,] 1.051331e-07 1.289639e-07 -2.842925e-08 -5.466062e-07 -6.219568e-07
## [21,] -6.675680e-08 2.463119e-07 -1.092586e-07 -2.596819e-06 -6.547574e-07
## [22,] 1.859184e-08 1.313031e-07 -5.740604e-08 -1.381563e-06 -2.594616e-07
## [23,] 1.496162e-07 1.250878e-07 -1.603083e-08 -6.245206e-08 2.164174e-07
## [24,] 8.613689e-08 2.255376e-07 -3.921166e-08 -3.324395e-07 5.192587e-08
## [25,] -6.942056e-09 1.152698e-07 -1.161996e-08 -1.895262e-07 5.233199e-09
## [26,] -9.430926e-07 -3.023790e-07 1.665966e-07 1.151382e-06 8.686536e-07
## [27,] -1.840999e-06 -4.929527e-07 -3.217512e-07 1.101964e-06 8.641482e-07
## [28,] -3.925516e-07 -1.086817e-06 1.955203e-07 7.521414e-07 5.590533e-07
## [29,] -3.272395e-07 1.901244e-07 -5.903691e-07 -1.573108e-07 -3.994475e-08
## [30,] 6.855576e-07 6.322135e-07 -1.602470e-07 -1.500402e-05 -3.334986e-06
## [31,] 7.562487e-07 3.215301e-07 3.162457e-08 -1.211704e-06 -1.108275e-05
## [32,] 4.055767e-07 3.998505e-08 7.584082e-08 1.978843e-06 -1.970608e-06
## [33,] 5.983249e-07 6.976673e-08 1.577699e-07 5.435712e-06 2.078081e-06
## [34,] 3.800437e-07 1.003800e-07 8.609202e-08 3.189851e-06 1.261379e-06
## [35,] 2.676613e-07 2.229780e-07 2.555938e-10 8.551447e-07 5.991060e-07
## [36,] 2.382349e-07 1.427327e-07 6.715513e-09 7.230974e-07 5.293449e-07
## [37,] 6.442480e-08 2.573549e-08 2.994976e-08 4.697526e-07 3.143642e-07
## [38,] -1.837701e-06 -5.782582e-07 3.386214e-07 2.425646e-06 1.773053e-06
## [39,] -3.495743e-06 -1.266774e-06 -4.090963e-07 3.603348e-06 1.808921e-06
## [40,] -9.532017e-07 -1.310331e-06 3.495864e-08 -1.548005e-06 1.009766e-06
## [41,] -5.260970e-07 -1.243038e-08 -7.357028e-07 1.501211e-06 -4.671475e-08
## [42,] 2.444701e-07 -3.559390e-06 1.947036e-06 4.042948e-05 6.814852e-07
## [43,] 1.278223e-06 -3.148425e-07 3.816061e-07 9.833220e-06 -1.042498e-05
## [44,] 3.585210e-07 3.971234e-07 -8.377871e-08 -1.927098e-06 -3.788115e-06
## [45,] -4.317852e-07 6.924829e-07 -3.969912e-07 -1.029014e-05 -2.829246e-06
## [46,] -6.341564e-08 3.656655e-07 -1.958629e-07 -5.038421e-06 -1.140386e-06
## [47,] 3.859161e-07 3.961479e-07 -4.908911e-08 -8.665338e-08 5.759170e-07
## [48,] 3.313809e-07 3.859590e-07 -4.169099e-08 1.723785e-07 5.400821e-07
## [49,] 5.492312e-08 1.530260e-07 2.392137e-08 2.698182e-07 3.608912e-07
## [50,] -3.956906e-06 -1.235237e-06 7.373083e-07 4.900000e-06 3.602323e-06
## [51,] 3.929425e-04 -2.160655e-06 -8.506514e-07 3.039225e-06 3.543906e-06
## [52,] -2.160655e-06 3.965856e-04 6.023115e-07 7.106430e-06 2.283535e-06
## [53,] -8.506514e-07 6.023115e-07 3.984928e-04 -3.029212e-06 -3.198640e-07
## [54,] 3.039225e-06 7.106430e-06 -3.029212e-06 2.863377e-04 -1.690354e-05
## [55,] 3.543906e-06 2.283535e-06 -3.198640e-07 -1.690354e-05 3.532619e-04
## [56,] 1.492912e-06 1.521377e-07 2.825644e-07 7.135233e-06 -8.826379e-06
## [57,] 2.061788e-06 2.643030e-07 6.088534e-07 2.116091e-05 7.677978e-06
## [58,] 1.229619e-06 4.097434e-07 2.939360e-07 1.131544e-05 4.266132e-06
## [59,] 7.585368e-07 6.831765e-07 -1.408938e-08 2.186861e-06 1.637977e-06
## [60,] 6.125318e-07 5.463727e-07 -2.453324e-08 1.300159e-06 1.225784e-06
## [61,] 1.233323e-07 1.960353e-07 7.429885e-08 1.006297e-06 8.329973e-07
## [62,] -8.283019e-06 -2.577130e-06 1.578490e-06 1.018218e-05 7.420535e-06
## [63,] -1.425352e-05 -5.677956e-06 -7.187341e-07 1.812084e-05 7.933165e-06
## [64,] -4.522794e-06 -3.390230e-06 -3.433997e-07 -1.389961e-05 3.057125e-06
## [65,] -1.453462e-06 -5.866457e-07 -1.736105e-06 1.039578e-05 1.547880e-07
## [66,] -1.595515e-05 3.625589e-05 -8.523298e-06 -1.145046e-04 -4.958553e-05
## [,56] [,57] [,58] [,59] [,60]
## [1,] -1.225256e-06 -5.018402e-06 -4.221252e-06 -2.988899e-06 -2.408519e-06
## [2,] 7.376936e-08 4.355426e-07 2.368078e-07 7.226641e-08 5.283806e-08
## [3,] -6.448106e-07 2.710203e-07 1.471119e-07 4.569579e-08 3.813737e-08
## [4,] -9.709896e-07 -1.173946e-06 -8.083826e-08 -5.569023e-08 -3.149796e-08
## [5,] -8.948254e-07 -3.798048e-06 -1.500448e-06 -1.748225e-07 -6.805998e-08
## [6,] 1.715754e-07 -1.273372e-06 -1.992729e-06 -1.150191e-06 -1.513708e-07
## [7,] 1.270496e-07 1.189811e-07 -1.002391e-06 -1.396940e-06 -3.140392e-07
## [8,] 1.555114e-07 -1.687243e-07 -2.957081e-07 -4.667180e-07 -1.624907e-06
## [9,] 9.432672e-08 -2.337995e-07 -2.035277e-07 -1.623801e-07 -5.619991e-07
## [10,] 1.873502e-07 2.206129e-08 -1.915303e-08 3.300283e-09 1.722302e-08
## [11,] 1.773287e-07 -1.756983e-07 -1.657271e-07 -7.666397e-08 -3.614804e-08
## [12,] 1.769495e-07 2.381651e-07 1.501298e-07 1.031346e-07 8.414379e-08
## [13,] -2.491334e-08 -1.051070e-07 -7.113805e-08 -3.468609e-08 -2.399877e-08
## [14,] -1.779394e-07 5.695182e-06 3.509972e-06 -7.035241e-07 -9.519382e-07
## [15,] -4.985554e-06 -8.866833e-06 -5.812330e-06 -3.224205e-06 -2.567351e-06
## [16,] -2.674165e-06 2.142850e-05 1.051319e-05 -3.900965e-06 -4.281018e-06
## [17,] -1.194133e-05 -2.470521e-05 -1.359908e-05 -4.631592e-06 -3.365966e-06
## [18,] -4.987145e-07 -8.684295e-07 -4.247013e-07 2.689912e-08 4.265312e-08
## [19,] -6.541308e-07 -5.301257e-07 -3.245104e-07 -1.628804e-08 2.008900e-08
## [20,] -5.997125e-08 4.309992e-07 -5.912401e-08 6.387016e-08 6.043807e-08
## [21,] 3.332958e-07 3.223392e-06 2.162261e-06 8.548765e-08 4.714146e-08
## [22,] -1.207692e-07 1.863611e-06 1.537983e-06 -5.458938e-07 -1.084068e-07
## [23,] 7.876975e-08 8.071581e-08 -4.691899e-07 -1.079924e-06 -2.511644e-07
## [24,] 3.409447e-08 3.376918e-07 1.719166e-07 -9.736516e-08 -1.161458e-06
## [25,] 1.057469e-08 2.228410e-07 1.839774e-07 1.036046e-07 -2.144264e-07
## [26,] 2.971640e-07 3.226996e-07 2.005198e-07 1.672678e-07 1.460785e-07
## [27,] 3.338230e-07 3.894950e-07 2.249367e-07 1.500677e-07 1.257839e-07
## [28,] 1.151858e-07 3.152214e-07 2.740641e-07 2.678429e-07 2.056810e-07
## [29,] 4.061419e-08 6.082253e-08 1.532146e-08 -2.796424e-08 -2.247028e-08
## [30,] 7.298997e-07 3.066483e-06 1.685436e-06 4.218915e-07 2.707661e-07
## [31,] -2.367687e-06 1.559633e-06 9.160042e-07 3.868689e-07 2.873282e-07
## [32,] -3.129932e-06 -3.155346e-06 3.299682e-07 1.737771e-07 1.256253e-07
## [33,] -3.194895e-06 -1.199785e-05 -4.120006e-06 3.340118e-07 3.290249e-07
## [34,] 3.502535e-07 -3.935477e-06 -5.316665e-06 -2.427022e-06 -6.664226e-08
## [35,] 2.421632e-07 6.372363e-07 -2.251115e-06 -3.276068e-06 -5.379706e-07
## [36,] 2.000907e-07 4.392825e-07 -1.939002e-08 -5.722497e-07 -3.324404e-06
## [37,] 1.119853e-07 1.426035e-07 8.983962e-08 -1.566573e-08 -9.826514e-07
## [38,] 6.127218e-07 7.787123e-07 4.882983e-07 3.652780e-07 3.055832e-07
## [39,] 5.357647e-07 5.183147e-07 3.265565e-07 3.066522e-07 2.612635e-07
## [40,] 5.772158e-07 1.350634e-06 8.865828e-07 5.035681e-07 3.725225e-07
## [41,] -1.380356e-07 -3.781089e-07 -2.287855e-07 -8.280607e-08 -5.568852e-08
## [42,] -3.924654e-06 -6.988689e-06 -3.636853e-06 -3.322107e-07 -9.998814e-08
## [43,] -3.970126e-06 -2.288495e-06 -1.444354e-06 -6.274099e-08 9.288262e-08
## [44,] 4.335252e-07 2.667117e-06 -2.027422e-07 1.835974e-07 1.832587e-07
## [45,] 2.150008e-06 1.511396e-05 8.684625e-06 -4.114046e-08 -3.928512e-08
## [46,] -4.520036e-07 8.285830e-06 5.937705e-06 -2.130388e-06 -4.151285e-07
## [47,] 2.185525e-07 6.829239e-07 -1.500440e-06 -3.470666e-06 -7.197897e-07
## [48,] 2.135529e-07 6.418158e-07 1.371251e-07 -5.903624e-07 -4.239116e-06
## [49,] 1.350443e-07 3.451654e-07 2.638460e-07 9.952836e-08 -1.053320e-06
## [50,] 1.238242e-06 1.537019e-06 9.636324e-07 7.356637e-07 6.204834e-07
## [51,] 1.492912e-06 2.061788e-06 1.229619e-06 7.585368e-07 6.125318e-07
## [52,] 1.521377e-07 2.643030e-07 4.097434e-07 6.831765e-07 5.463727e-07
## [53,] 2.825644e-07 6.088534e-07 2.939360e-07 -1.408938e-08 -2.453324e-08
## [54,] 7.135233e-06 2.116091e-05 1.131544e-05 2.186861e-06 1.300159e-06
## [55,] -8.826379e-06 7.677978e-06 4.266132e-06 1.637977e-06 1.225784e-06
## [56,] 3.892620e-04 -1.099169e-05 1.159621e-06 6.586580e-07 4.737886e-07
## [57,] -1.099169e-05 3.539400e-04 -1.663712e-05 1.628528e-06 1.351195e-06
## [58,] 1.159621e-06 -1.663712e-05 3.816944e-04 -6.327795e-06 7.027673e-08
## [59,] 6.586580e-07 1.628528e-06 -6.327795e-06 3.908538e-04 -1.380250e-06
## [60,] 4.737886e-07 1.351195e-06 7.027673e-08 -1.380250e-06 3.913876e-04
## [61,] 3.044276e-07 6.348103e-07 4.498407e-07 1.145545e-07 -2.405303e-06
## [62,] 2.551379e-06 3.221929e-06 2.016932e-06 1.514745e-06 1.268354e-06
## [63,] 2.024426e-06 1.945005e-06 1.319512e-06 1.391198e-06 1.175185e-06
## [64,] 2.802739e-06 5.971415e-06 3.650888e-06 1.689481e-06 1.231728e-06
## [65,] -9.392371e-07 -2.094561e-06 -1.163144e-06 -2.846442e-07 -1.805782e-07
## [66,] -1.236093e-05 9.697514e-05 7.742444e-05 4.605459e-05 3.035617e-05
## [,61] [,62] [,63] [,64] [,65]
## [1,] -9.087086e-07 -4.699933e-07 -3.936609e-07 -1.870912e-06 -2.623988e-07
## [2,] 3.642180e-08 3.232340e-07 5.705443e-07 -4.238568e-07 3.282216e-07
## [3,] 3.189334e-08 3.296436e-07 2.722229e-07 2.946487e-07 -1.039126e-07
## [4,] 4.311014e-09 2.646690e-07 2.192270e-07 1.478002e-07 -8.065800e-08
## [5,] 1.152071e-09 4.230983e-07 3.274238e-07 2.671780e-07 -1.634222e-07
## [6,] 1.002863e-08 2.939104e-07 2.271826e-07 2.132891e-07 -1.137247e-07
## [7,] -4.275814e-09 2.455000e-07 2.191335e-07 1.846603e-07 -5.225543e-08
## [8,] -4.882826e-07 2.660183e-07 2.300484e-07 5.536494e-08 -6.269617e-08
## [9,] -3.025212e-07 -1.575100e-07 9.387797e-08 -3.538134e-08 -2.644769e-08
## [10,] -2.901002e-07 -1.871614e-06 -9.293262e-07 -2.251565e-07 7.304006e-08
## [11,] -3.438891e-08 -9.055465e-07 -1.835181e-06 -4.835596e-07 -4.107966e-07
## [12,] 3.893198e-08 -6.087623e-08 -2.615107e-07 -1.002105e-06 1.344995e-07
## [13,] 1.232655e-09 1.263225e-07 -3.499150e-07 1.484781e-07 -5.750123e-07
## [14,] -5.381691e-07 -4.557152e-06 -4.848405e-06 -2.461889e-06 -3.900716e-07
## [15,] -1.279244e-06 -5.646722e-06 -5.345635e-06 -4.270122e-06 3.011435e-07
## [16,] -2.816439e-06 -2.404866e-05 -2.820111e-05 -4.957648e-06 -4.255576e-06
## [17,] -2.423329e-06 -2.317753e-05 -1.903181e-05 -2.326876e-05 7.821308e-06
## [18,] 7.060323e-09 -1.407543e-07 -2.593597e-07 3.029971e-07 -1.505735e-07
## [19,] 3.427589e-08 4.934026e-07 4.421804e-07 3.388137e-07 -1.051859e-07
## [20,] 3.434482e-08 2.454825e-07 2.795615e-07 9.307792e-08 3.258090e-08
## [21,] 1.956972e-08 1.759972e-08 1.741887e-07 -2.830533e-07 2.265740e-07
## [22,] 1.261988e-08 1.144554e-07 2.064756e-07 -1.538591e-07 1.261473e-07
## [23,] 1.396718e-08 3.163913e-07 3.302886e-07 1.830531e-07 -1.008803e-09
## [24,] -2.387686e-07 2.278124e-07 2.360548e-07 2.777176e-07 3.991976e-09
## [25,] -1.274245e-07 -4.364556e-07 3.536161e-08 1.376160e-07 2.015816e-08
## [26,] -4.547424e-07 -3.639177e-06 -1.908531e-06 -3.526316e-07 1.772738e-07
## [27,] 6.243348e-09 -1.855682e-06 -3.519433e-06 -1.014600e-06 -5.789598e-07
## [28,] 7.689765e-08 -4.105976e-07 -1.079069e-06 -1.219440e-06 -3.266018e-08
## [29,] 1.337061e-08 3.241082e-07 -4.183381e-07 1.316605e-08 -7.313694e-07
## [30,] 1.797335e-07 1.572402e-06 2.802624e-06 -2.063006e-06 1.615590e-06
## [31,] 1.792454e-07 1.446178e-06 1.372505e-06 1.076587e-06 -2.148892e-07
## [32,] 8.187929e-08 6.951382e-07 5.562663e-07 7.442066e-07 -2.482360e-07
## [33,] 1.659562e-07 9.622635e-07 6.438341e-07 1.534439e-06 -5.315536e-07
## [34,] 1.246521e-07 6.313910e-07 4.402311e-07 1.004665e-06 -3.223687e-07
## [35,] 3.320255e-08 5.283783e-07 4.795833e-07 5.894727e-07 -1.062814e-07
## [36,] -9.545774e-07 4.708211e-07 4.264268e-07 4.236327e-07 -8.137951e-08
## [37,] -5.713729e-07 -8.099735e-07 1.020723e-07 1.615463e-07 -8.717895e-09
## [38,] -8.970551e-07 -7.532276e-06 -4.140365e-06 -7.536374e-07 3.991930e-07
## [39,] 1.930788e-08 -4.122888e-06 -7.425711e-06 -1.808145e-06 -1.171072e-06
## [40,] 1.550770e-07 -7.109563e-07 -1.736764e-06 -2.889926e-06 3.912976e-07
## [41,] 1.122687e-08 5.097315e-07 -1.052652e-06 4.232365e-07 -1.455154e-06
## [42,] -1.553834e-07 -2.219269e-06 -4.324336e-06 4.208172e-06 -2.775708e-06
## [43,] 1.427460e-07 1.983805e-06 1.346999e-06 2.411112e-06 -1.030571e-06
## [44,] 1.083718e-07 8.186536e-07 9.167408e-07 3.258134e-07 8.561463e-08
## [45,] -1.967593e-08 -3.068923e-07 3.165582e-07 -1.576663e-06 9.299378e-07
## [46,] 6.349848e-09 1.422012e-07 4.687400e-07 -7.380616e-07 4.661840e-07
## [47,] 4.030100e-08 8.264272e-07 8.490124e-07 6.158478e-07 -2.451638e-08
## [48,] -1.060147e-06 7.241468e-07 7.041364e-07 6.698950e-07 -5.607772e-08
## [49,] -7.063822e-07 -1.607804e-06 1.453275e-07 3.137329e-07 2.721442e-08
## [50,] -1.655201e-06 -1.515600e-05 -8.704487e-06 -1.619122e-06 8.887222e-07
## [51,] 1.233323e-07 -8.283019e-06 -1.425352e-05 -4.522794e-06 -1.453462e-06
## [52,] 1.960353e-07 -2.577130e-06 -5.677956e-06 -3.390230e-06 -5.866457e-07
## [53,] 7.429885e-08 1.578490e-06 -7.187341e-07 -3.433997e-07 -1.736105e-06
## [54,] 1.006297e-06 1.018218e-05 1.812084e-05 -1.389961e-05 1.039578e-05
## [55,] 8.329973e-07 7.420535e-06 7.933165e-06 3.057125e-06 1.547880e-07
## [56,] 3.044276e-07 2.551379e-06 2.024426e-06 2.802739e-06 -9.392371e-07
## [57,] 6.348103e-07 3.221929e-06 1.945005e-06 5.971415e-06 -2.094561e-06
## [58,] 4.498407e-07 2.016932e-06 1.319512e-06 3.650888e-06 -1.163144e-06
## [59,] 1.145545e-07 1.514745e-06 1.391198e-06 1.689481e-06 -2.846442e-07
## [60,] -2.405303e-06 1.268354e-06 1.175185e-06 1.231728e-06 -1.805782e-07
## [61,] 3.983826e-04 -3.121018e-06 2.429420e-07 5.710472e-07 1.983967e-08
## [62,] -3.121018e-06 3.689960e-04 -1.860859e-05 -3.505450e-06 1.982417e-06
## [63,] 2.429420e-07 -1.860859e-05 3.689416e-04 -7.585466e-06 -3.461285e-06
## [64,] 5.710472e-07 -3.505450e-06 -7.585466e-06 3.897564e-04 1.666173e-06
## [65,] 1.983967e-08 1.982417e-06 -3.461285e-06 1.666173e-06 3.954000e-04
## [66,] 9.249532e-06 -1.997825e-05 -1.765605e-05 3.000705e-05 4.186419e-06
## [,66]
## [1,] -1.260574e-04
## [2,] -3.626361e-06
## [3,] -6.311082e-06
## [4,] -1.493025e-05
## [5,] -2.623247e-05
## [6,] -1.526203e-05
## [7,] -4.300049e-06
## [8,] -2.218876e-05
## [9,] -1.939114e-05
## [10,] -1.241618e-05
## [11,] -1.998392e-05
## [12,] -1.397648e-06
## [13,] -2.320375e-06
## [14,] 5.701084e-05
## [15,] 8.019525e-05
## [16,] 9.052213e-05
## [17,] 1.570192e-04
## [18,] 1.017970e-05
## [19,] -1.238721e-05
## [20,] 1.297193e-06
## [21,] 7.913723e-06
## [22,] -1.554536e-06
## [23,] 9.969276e-07
## [24,] 1.880251e-05
## [25,] 1.202901e-05
## [26,] -3.574090e-06
## [27,] -6.212121e-06
## [28,] 1.501326e-05
## [29,] -3.646391e-06
## [30,] -8.109113e-06
## [31,] -3.141515e-06
## [32,] -4.360917e-06
## [33,] 1.755875e-05
## [34,] 1.581941e-05
## [35,] 1.449813e-05
## [36,] 4.450066e-06
## [37,] -2.015044e-06
## [38,] -4.769992e-06
## [39,] -8.082144e-06
## [40,] 1.737034e-05
## [41,] -3.047119e-06
## [42,] 5.023227e-05
## [43,] -4.959046e-05
## [44,] 2.045903e-06
## [45,] 9.203763e-06
## [46,] -7.463512e-06
## [47,] 1.460566e-05
## [48,] 2.247994e-05
## [49,] 9.804547e-06
## [50,] -9.934329e-06
## [51,] -1.595515e-05
## [52,] 3.625589e-05
## [53,] -8.523298e-06
## [54,] -1.145046e-04
## [55,] -4.958553e-05
## [56,] -1.236093e-05
## [57,] 9.697514e-05
## [58,] 7.742444e-05
## [59,] 4.605459e-05
## [60,] 3.035617e-05
## [61,] 9.249532e-06
## [62,] -1.997825e-05
## [63,] -1.765605e-05
## [64,] 3.000705e-05
## [65,] 4.186419e-06
## [66,] 5.400261e-03
##
## $log_evidence
## [1] -131.93
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 134
Use the viz_post_coefs() function to visualize
the posterior coefficient summaries for model 3 and model 6, based on
the very strong prior specification.
post_means_3_2 <- laplace_03_very_strong$mode[1:ncol(X03)]
post_sds_3_2 <- sqrt(diag(laplace_03_very_strong$var_matrix)[1:ncol(X03)])
viz_post_coefs(post_means_3_2, post_sds_3_2, xnames_3)
post_means_6_2 <- laplace_06_very_strong$mode[1:ncol(X06)]
post_sds_6_2 <- sqrt(diag(laplace_06_very_strong$var_matrix)[1:ncol(X06)])
viz_post_coefs(post_means_6_2, post_sds_6_2, xnames_6)
Describe the influence of the regression coefficient prior standard deviation on the coefficient posterior distributions.
What do you think?
The prior standard deviation of regression coefficients affects how much
the prior information influences the coefficient posterior
distributions. A diffuse or weak prior with a larger standard deviation
provides a wider range of possible values for the coefficients, whereas
a strong or informative prior with a smaller standard deviation
constrains the range of possible values, giving more weight to the prior
information. This leads to more narrowly distributed and less variable
coefficient posterior distributions with a strong prior, indicating a
lower chance of overfitting to the data. Hence, the prior standard
deviation plays a crucial role in balancing the influence of the prior
knowledge and the data in the model fitting process.
You previously compared the two models using the Bayes Factor based on the weak prior specification.
Compare the performance of the two models with Bayes Factors again, but considering the results based on the strong and very strong priors. Does the prior influence which model is considered to be better?
bf1 <- exp(laplace_03_strong$log_evidence - laplace_06_strong$log_evidence)
bf1
## [1] 2.672262e+13
bf1 is greater than 1, then Model 3 is favored over Model 6.
bf2 <- exp(laplace_03_very_strong$log_evidence - laplace_06_very_strong$log_evidence)
bf2
## [1] 0.0002438238
bf2 is smaller than 1, then Model 6 is favored over Model 3.
You examined the behavior of the coefficient posterior based on the influence of the prior. Let’s now consider the prior’s influence by examining the posterior predictive distributions.
You will make posterior predictions following the approach from the previous assignment. Posterior samples are generated and those samples are used to calculate the posterior samples of the mean trend and generate random posterior samples of the response around the mean. In the previous assignment, you made posterior predictions in order to calculate errors. In this assignment, you will not calculate errors, instead you will summarize the posterior predictions of the mean and of the random response.
The generate_lm_post_samples() function is defined for
you below. It uses the MASS::mvrnorm() function generate
posterior samples from the Laplace Approximation’s MVN distribution.
generate_lm_post_samples <- function(mvn_result, length_beta, num_samples)
{
MASS::mvrnorm(n = num_samples,
mu = mvn_result$mode,
Sigma = mvn_result$var_matrix) %>%
as.data.frame() %>% tibble::as_tibble() %>%
purrr::set_names(c(sprintf("beta_%02d", 0:(length_beta-1)), "varphi")) %>%
mutate(sigma = exp(varphi))
}
The code chunk below starts the post_lm_pred_samples()
function. This function generates posterior mean trend predictions and
posterior predictions of the response. The first argument,
Xnew, is a potentially new or test design matrix that we
wish to make predictions at. The second argument, Bmat, is
a matrix of posterior samples of the \(\boldsymbol{\beta}\)-parameters, and the
third argument, sigma_vector, is a vector of posterior
samples of the likelihood noise. The Xnew matrix has rows
equal to the number of predictions points, M, and the
Bmat matrix has rows equal to the number of posterior
samples S.
You must complete the function by performing the necessary matrix
math to calculate the matrix of posterior mean trend predictions,
Umat, and the matrix of posterior response predictions,
Ymat. You must also complete missing arguments to the
definition of the Rmat and Zmat matrices. The
Rmat matrix replicates the posterior likelihood noise
samples the correct number of times. The Zmat matrix is the
matrix of randomly generated standard normal values. You must correctly
specify the required number of rows to the Rmat and
Zmat matrices.
The post_lm_pred_samples() returns the Umat
and Ymat matrices contained within a list.
Perform the necessary matrix math to calculate the matrix of
posterior predicted mean trends Umat and posterior
predicted responses Ymat. You must specify the number of
required rows to create the Rmat and Zmat
matrices.
HINT: The following code chunk should look famaliar…
post_lm_pred_samples <- function(Xnew, Bmat, sigma_vector)
{
# number of new prediction locations
M <- nrow(Xnew)
# number of posterior samples
S <- nrow(Bmat)
# matrix of linear predictors
Umat <- Xnew %*% t(Bmat)
# assmeble matrix of sigma samples, set the number of rows
Rmat <- matrix(rep(sigma_vector, M), M, byrow = TRUE)
# generate standard normal and assemble into matrix
# set the number of rows
Zmat <- matrix(rnorm(M*S), M, byrow = TRUE)
# calculate the random observation predictions
Ymat <- Umat + Rmat * Zmat
# package together
list(Umat = Umat, Ymat = Ymat)
}
Since this assignment is focused on visualizing the predictions, we
will summarize the posterior predictions to focus on the posterior means
and the middle 95% uncertainty intervals. The code chunk below is
defined for you which serves as a useful wrapper function to call
post_lm_pred_samples().
make_post_lm_pred <- function(Xnew, post)
{
Bmat <- post %>% select(starts_with("beta_")) %>% as.matrix()
sigma_vector <- post %>% pull(sigma)
post_lm_pred_samples(Xnew, Bmat, sigma_vector)
}
The code chunk below defines a function
summarize_lm_pred_from_laplace() which manages the actions
necessary to summarize posterior predictions. The first argument,
mvn_result, is the Laplace Approximation object. The second
object is the test design matrix, Xtest, and the third
argument, num_samples, is the number of posterior samples
to make.
You must complete the code chunk below which summarizes the posterior
predictions. This function takes care of most of the coding for you. You
do not have to worry about the generation of the posterior samples OR
calculating the posterior quantiles associated with the middle 95%
uncertainty interval. You must calculate the posterior average by
deciding on whether you should use colMeans() or
rowMeans() to calculate the average across all posterior
samples per prediction location.
Follow the comments in the code chunk below to complete the definition of the summarize_lm_pred_from_laplace() function. You must calculate the average posterior mean trend and the average posterior response.
summarize_lm_pred_from_laplace <- function(mvn_result, Xtest, num_samples)
{
# generate posterior samples of the beta parameters
post <- generate_lm_post_samples(mvn_result, ncol(Xtest), num_samples)
# make posterior predictions on the test set
pred_test <- make_post_lm_pred(Xtest, post)
# calculate summary statistics on the predicted mean and response
# summarize over the posterior samples
# posterior mean, should you summarize along rows (rowMeans) or
# summarize down columns (colMeans) ???
mu_avg <- rowMeans(pred_test$Umat)
y_avg <- rowMeans(pred_test$Umat)
# posterior quantiles for the middle 95% uncertainty intervals
mu_lwr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.025)
mu_upr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.975)
y_lwr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.025)
y_upr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.975)
# book keeping
tibble::tibble(
mu_avg = mu_avg,
mu_lwr = mu_lwr,
mu_upr = mu_upr,
y_avg = y_avg,
y_lwr = y_lwr,
y_upr = y_upr
) %>%
tibble::rowid_to_column("pred_id")
}
When you made predictions in Problem 02, the lm() object
handled making the test design matrix. However, since we have programmed
the Bayesian modeling approach from scratch we need to create the test
design matrix manually.
Create the test design matrix based on the visualization
grid, viz_grid, using the model 3 formulation. Assign the
result to the X03_test object.
Call the summarize_lm_pred_from_laplace()
function to summarize the posterior predictions from the model 3
formulation for the weak, strong, and very strong prior specifications.
Use 5000 posterior samples for each case. Assign the results from the
weak prior to post_pred_summary_viz_03_weak, the results
from the strong prior to post_pred_summary_viz_03_strong,
and the results from the very strong prior to
post_pred_summary_viz_03_very_strong.
X03_test <- model.matrix( ~ (x1 + I(x1^2)) * (x2 + I(x2^2)), viz_grid)
post_pred_summary_viz_03_weak <- summarize_lm_pred_from_laplace(laplace_03_weak,X03_test, 5000)
post_pred_summary_viz_03_weak
## # A tibble: 909 × 7
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -1.13 -9.60 7.47 -1.13 -9.59 7.55
## 2 2 -1.19 -9.26 7.07 -1.19 -9.37 7.17
## 3 3 -1.25 -8.93 6.65 -1.25 -8.96 6.76
## 4 4 -1.31 -8.64 6.21 -1.31 -8.75 6.24
## 5 5 -1.37 -8.36 5.75 -1.37 -8.52 5.90
## 6 6 -1.44 -8.08 5.31 -1.44 -8.19 5.50
## 7 7 -1.50 -7.78 4.90 -1.50 -8.02 5.15
## 8 8 -1.56 -7.51 4.51 -1.56 -7.68 4.61
## 9 9 -1.62 -7.28 4.16 -1.62 -7.64 4.27
## 10 10 -1.68 -7.07 3.79 -1.68 -7.25 3.93
## # … with 899 more rows
post_pred_summary_viz_03_strong <- summarize_lm_pred_from_laplace(laplace_03_strong,X03_test, 5000)
post_pred_summary_viz_03_strong
## # A tibble: 909 × 7
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -1.21 -9.59 6.85 -1.21 -9.81 6.97
## 2 2 -1.27 -9.30 6.44 -1.27 -9.26 6.50
## 3 3 -1.33 -9.00 6.03 -1.33 -9.15 6.18
## 4 4 -1.39 -8.70 5.63 -1.39 -8.93 5.83
## 5 5 -1.45 -8.40 5.24 -1.45 -8.43 5.37
## 6 6 -1.51 -8.16 4.87 -1.51 -8.20 4.97
## 7 7 -1.57 -7.87 4.49 -1.57 -8.03 4.68
## 8 8 -1.63 -7.63 4.11 -1.63 -7.73 4.24
## 9 9 -1.69 -7.35 3.75 -1.69 -7.55 4.00
## 10 10 -1.75 -7.13 3.39 -1.75 -7.31 3.60
## # … with 899 more rows
post_pred_summary_viz_03_very_strong <- summarize_lm_pred_from_laplace(laplace_03_very_strong,X03_test, 5000)
post_pred_summary_viz_03_very_strong
## # A tibble: 909 × 7
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -3.59 -7.07 -0.103 -3.59 -7.52 0.431
## 2 2 -3.47 -6.82 -0.110 -3.47 -7.29 0.441
## 3 3 -3.34 -6.56 -0.120 -3.34 -6.94 0.511
## 4 4 -3.22 -6.31 -0.121 -3.22 -6.75 0.311
## 5 5 -3.10 -6.06 -0.119 -3.10 -6.70 0.338
## 6 6 -2.99 -5.83 -0.125 -2.99 -6.24 0.319
## 7 7 -2.88 -5.59 -0.124 -2.88 -6.14 0.387
## 8 8 -2.76 -5.37 -0.127 -2.76 -5.96 0.402
## 9 9 -2.66 -5.15 -0.139 -2.66 -5.68 0.495
## 10 10 -2.55 -4.93 -0.144 -2.55 -5.42 0.436
## # … with 899 more rows
You will now visualize the posterior predictions from the model 3
Bayesian models associated with the weak, strong, and very strong
priors. The viz_grid object is joined to the prediction
dataframes assuming you have used the correct variable names!
Visualize the predicted means, confidence intervals, and
prediction intervals in the style of those that you created in Problem
02. The confidence interval bounds are mu_lwr and
mu_upr columns and the prediction interval bounds are the
y_lwr and y_upr columns, respectively. The
posterior predicted mean of the mean is
mu_avg.
Pipe the result of the joined dataframe into
ggplot() and make appropriate aesthetics and layers to
visualize the predictions with the x1 variable mapped to
the x aesthetic and the x2 variable used as a
facet variable.
post_pred_summary_viz_03_weak %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')
## # A tibble: 909 × 9
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr x1 x2
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -1.13 -9.60 7.47 -1.13 -9.59 7.55 -3.2 -3
## 2 2 -1.19 -9.26 7.07 -1.19 -9.37 7.17 -3.14 -3
## 3 3 -1.25 -8.93 6.65 -1.25 -8.96 6.76 -3.07 -3
## 4 4 -1.31 -8.64 6.21 -1.31 -8.75 6.24 -3.01 -3
## 5 5 -1.37 -8.36 5.75 -1.37 -8.52 5.90 -2.94 -3
## 6 6 -1.44 -8.08 5.31 -1.44 -8.19 5.50 -2.88 -3
## 7 7 -1.50 -7.78 4.90 -1.50 -8.02 5.15 -2.82 -3
## 8 8 -1.56 -7.51 4.51 -1.56 -7.68 4.61 -2.75 -3
## 9 9 -1.62 -7.28 4.16 -1.62 -7.64 4.27 -2.69 -3
## 10 10 -1.68 -7.07 3.79 -1.68 -7.25 3.93 -2.62 -3
## # … with 899 more rows
post_pred_summary_viz_03_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')
## # A tibble: 909 × 9
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr x1 x2
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -1.21 -9.59 6.85 -1.21 -9.81 6.97 -3.2 -3
## 2 2 -1.27 -9.30 6.44 -1.27 -9.26 6.50 -3.14 -3
## 3 3 -1.33 -9.00 6.03 -1.33 -9.15 6.18 -3.07 -3
## 4 4 -1.39 -8.70 5.63 -1.39 -8.93 5.83 -3.01 -3
## 5 5 -1.45 -8.40 5.24 -1.45 -8.43 5.37 -2.94 -3
## 6 6 -1.51 -8.16 4.87 -1.51 -8.20 4.97 -2.88 -3
## 7 7 -1.57 -7.87 4.49 -1.57 -8.03 4.68 -2.82 -3
## 8 8 -1.63 -7.63 4.11 -1.63 -7.73 4.24 -2.75 -3
## 9 9 -1.69 -7.35 3.75 -1.69 -7.55 4.00 -2.69 -3
## 10 10 -1.75 -7.13 3.39 -1.75 -7.31 3.60 -2.62 -3
## # … with 899 more rows
post_pred_summary_viz_03_very_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')
## # A tibble: 909 × 9
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr x1 x2
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -3.59 -7.07 -0.103 -3.59 -7.52 0.431 -3.2 -3
## 2 2 -3.47 -6.82 -0.110 -3.47 -7.29 0.441 -3.14 -3
## 3 3 -3.34 -6.56 -0.120 -3.34 -6.94 0.511 -3.07 -3
## 4 4 -3.22 -6.31 -0.121 -3.22 -6.75 0.311 -3.01 -3
## 5 5 -3.10 -6.06 -0.119 -3.10 -6.70 0.338 -2.94 -3
## 6 6 -2.99 -5.83 -0.125 -2.99 -6.24 0.319 -2.88 -3
## 7 7 -2.88 -5.59 -0.124 -2.88 -6.14 0.387 -2.82 -3
## 8 8 -2.76 -5.37 -0.127 -2.76 -5.96 0.402 -2.75 -3
## 9 9 -2.66 -5.15 -0.139 -2.66 -5.68 0.495 -2.69 -3
## 10 10 -2.55 -4.93 -0.144 -2.55 -5.42 0.436 -2.62 -3
## # … with 899 more rows
post_pred_summary_viz_03_weak %>%
right_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id') %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr,
ymax = y_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr,
ymax = mu_upr),
fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
coord_cartesian(ylim = c(-7, 7)) +
facet_wrap(~x2)
post_pred_summary_viz_03_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')%>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr,
ymax = y_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr,
ymax = mu_upr),
fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
coord_cartesian(ylim = c(-7, 7)) +
facet_wrap(~x2)
post_pred_summary_viz_03_very_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')%>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr,
ymax = y_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr,
ymax = mu_upr),
fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
coord_cartesian(ylim = c(-7, 7)) +
facet_wrap(~x2)
### 4e)
In order to make posterior predictions for the model 6 formulation
you must create a test design matrix consistent with the training set
basis. The code chunk below creates a helper function which extracts the
interior and boundary knots of a natural spline associated with the
training set for you. The first argument, J, is the
degrees-of-freedom (DOF) of the spline, the second argument,
train_data, is the training data set. The third argument
xname is the name of the variable you are applying the
spline to. The xname argument must be
provided as a character string.
make_splines_training_knots <- function(J, train_data, xname)
{
# extract the input from the training set
x <- train_data %>% select(all_of(xname)) %>% pull()
# create the training basis
train_basis <- splines::ns(x, df = J)
# extract the knots
interior_knots <- as.vector(attributes(train_basis)$knots)
boundary_knots <- as.vector(attributes(train_basis)$Boundary.knots)
# book keeping
list(interior_knots = interior_knots,
boundary_knots = boundary_knots)
}
Create the test design matrix based on the visualization
grid, viz_grid, using the model 6 formulation. Assign the
result to the X06_test object. Use the
make_splines_training_knots() function to get the interior
and boundary knots associated with the training set for the
x1 variable to create the test design matrix.
Call the summarize_lm_pred_from_laplace()
function to summarize the posterior predictions from the model 6
formulation for the weak, strong, and very strong prior specifications.
Use 5000 posterior samples for each case. Assign the results from the
weak prior to post_pred_summary_viz_06_weak, the results
from the strong prior to post_pred_summary_viz_06_strong,
and the results from the very strong prior to
post_pred_summary_viz_06_very_strong.
HINT: The make_spline_training_knots() function
returns a list! The fields or elements of the list can be accessed via
the $ operator.
# create test design matrix based on visualization grid for model 6
x1_knots <- make_splines_training_knots( 12, df, "x1")
X06_test <- model.matrix(~ splines::ns(x1, knots = x1_knots$interior_knots, Boundary.knots = x1_knots$boundary_knots) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), viz_grid)
# summarize posterior predictions for weak, strong, and very strong priors
post_pred_summary_viz_06_weak <- summarize_lm_pred_from_laplace(laplace_06_weak, X06_test, 5000)
post_pred_summary_viz_06_strong <- summarize_lm_pred_from_laplace(laplace_06_strong, X06_test, 5000)
post_pred_summary_viz_06_very_strong <- summarize_lm_pred_from_laplace(laplace_06_very_strong, X06_test, 5000)
You will now visualize the posterior predictions from the model 6
Bayesian models associated with the weak, strong, and very strong
priors. The viz_grid object is joined to the prediction
dataframes assuming you have used the correct variable names!
Visualize the predicted means, confidence intervals, and
prediction intervals in the style of those that you created in Problem
02. The confidence interval bounds are mu_lwr and
mu_upr columns and the prediction interval bounds are the
y_lwr and y_upr columns, respectively. The
posterior predicted mean of the mean is
mu_avg.
Pipe the result of the joined dataframe into
ggplot() and make appropriate aesthetics and layers to
visualize the predictions with the x1 variable mapped to
the x aesthetic and the x2 variable used as a
facet variable.
post_pred_summary_viz_06_weak %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')
## # A tibble: 909 × 9
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr x1 x2
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -446. -1977. 1050. -446. -1977. 1050. -3.2 -3
## 2 2 -422. -1868. 989. -422. -1868. 988. -3.14 -3
## 3 3 -398. -1762. 930. -398. -1762. 929. -3.07 -3
## 4 4 -373. -1653. 872. -373. -1653. 871. -3.01 -3
## 5 5 -349. -1546. 815. -349. -1546. 815. -2.94 -3
## 6 6 -325. -1439. 753. -325. -1438. 753. -2.88 -3
## 7 7 -301. -1332. 699. -301. -1333. 699. -2.82 -3
## 8 8 -277. -1224. 647. -277. -1225. 647. -2.75 -3
## 9 9 -252. -1116. 587. -252. -1115. 587. -2.69 -3
## 10 10 -228. -1008. 531. -228. -1007. 531. -2.62 -3
## # … with 899 more rows
post_pred_summary_viz_06_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')
## # A tibble: 909 × 9
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr x1 x2
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -36.3 -125. 53.5 -36.3 -126. 53.8 -3.2 -3
## 2 2 -34.7 -118. 49.8 -34.7 -118. 49.7 -3.14 -3
## 3 3 -33.1 -111. 45.4 -33.1 -111. 45.6 -3.07 -3
## 4 4 -31.5 -104. 41.8 -31.5 -104. 42.0 -3.01 -3
## 5 5 -29.9 -96.8 38.3 -29.9 -97.2 38.1 -2.94 -3
## 6 6 -28.3 -90.2 34.3 -28.3 -90.0 33.7 -2.88 -3
## 7 7 -26.7 -83.3 30.2 -26.7 -83.1 30.2 -2.82 -3
## 8 8 -25.1 -77.0 26.1 -25.1 -76.9 26.4 -2.75 -3
## 9 9 -23.5 -70.3 22.7 -23.5 -70.4 22.6 -2.69 -3
## 10 10 -21.8 -64.0 19.4 -21.8 -63.9 19.5 -2.62 -3
## # … with 899 more rows
post_pred_summary_viz_06_very_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')
## # A tibble: 909 × 9
## pred_id mu_avg mu_lwr mu_upr y_avg y_lwr y_upr x1 x2
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 -3.66 -7.22 -0.0909 -3.66 -7.60 0.312 -3.2 -3
## 2 2 -3.68 -7.04 -0.312 -3.68 -7.47 0.0557 -3.14 -3
## 3 3 -3.70 -6.88 -0.535 -3.70 -7.28 -0.124 -3.07 -3
## 4 4 -3.72 -6.75 -0.750 -3.72 -7.29 -0.397 -3.01 -3
## 5 5 -3.74 -6.60 -0.923 -3.74 -6.98 -0.470 -2.94 -3
## 6 6 -3.76 -6.46 -1.09 -3.76 -6.92 -0.613 -2.88 -3
## 7 7 -3.78 -6.30 -1.26 -3.78 -6.79 -0.823 -2.82 -3
## 8 8 -3.80 -6.18 -1.42 -3.80 -6.65 -1.03 -2.75 -3
## 9 9 -3.82 -6.09 -1.58 -3.82 -6.53 -1.02 -2.69 -3
## 10 10 -3.84 -6.03 -1.70 -3.84 -6.48 -1.15 -2.62 -3
## # … with 899 more rows
post_pred_summary_viz_06_weak %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')%>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr,
ymax = y_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr,
ymax = mu_upr),
fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
coord_cartesian(ylim = c(-7, 7)) +
facet_wrap(~x2)
post_pred_summary_viz_06_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')%>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr,
ymax = y_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr,
ymax = mu_upr),
fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
coord_cartesian(ylim = c(-7, 7)) +
facet_wrap(~x2)
post_pred_summary_viz_06_very_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id')%>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr,
ymax = y_upr),
fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr,
ymax = mu_upr),
fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
coord_cartesian(ylim = c(-7, 7)) +
facet_wrap(~x2)
Describe the behavior of the predictions as the prior standard deviation decreased. Are the posterior predictions consistent with the behavior of the posterior coefficients?
What do you think?
As the prior standard deviation gets smaller, we become more certain
about the posterior distribution of the coefficients. This increased
certainty in the coefficients results in more concentrated posterior
predictions around the observed values.
The reason for this is that a smaller prior standard deviation places more emphasis on the observed data, which leads to more confidence in the estimates of the coefficients. This, in turn, leads to more confidence in the predictions based on these coefficients.
Overall, the behavior of the posterior predictions is consistent with the behavior of the posterior coefficients. As we become more certain about the coefficients, we also become more certain about the predictions that are based on these coefficients.
Now that you have worked with Bayesian models with the prior
regularizing the coefficients, you will consider non-Bayesian
regularization methods. You will work with the glmnet
package in this problem which takes care of all fitting and
visualization for you.
The code chunk below loads in glmnet and so you must
have glmnet installed before running this code chunk.
IMPORANT: the eval flag is set to FALSE
below. Once you download glmnet set
eval=TRUE.
library(glmnet)
## Loading required package: Matrix
##
## Attaching package: 'Matrix'
## The following objects are masked from 'package:tidyr':
##
## expand, pack, unpack
## Loaded glmnet 4.1-7
glmnet does not work with the formula interface. And so
you must create the training design matrix. However, glmnet
prefers the the intercept column of ones to not be
included in the design matrix. To support that you must define new
design matrices. These matrices will use the same formulation but you
must remove the intercept column. This is easy to do with the formula
interface and the model.matrix() function. Include
- 1 in the formula and model.matrix() will not
include the intercept. The code chunk below demonstrates removing the
intercept column for a model with linear additive features.
model.matrix( y ~ x1 + x2 - 1, data = df) %>% head()
## x1 x2
## 1 -0.3092328 0.3087799
## 2 0.6312721 -0.5479198
## 3 -0.6827669 2.1664494
## 4 0.2693056 1.2097037
## 5 0.3725202 0.7854860
## 6 1.2966439 -0.1877231
Create the design matrices for glmnet for the
model 3 and model 6 formulations. Remove the intercept column for both
and assign the results to X03_glmnet and
X06_glmnet.
X03_glmnet <- model.matrix(y ~ (x1 + I(x1^2)) * (x2 + I(x2^2)) - 1, df)
X06_glmnet <- model.matrix(y ~ splines::ns(x1, 12) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)) - 1, df)
By default glmnet uses the lasso
penalty. Fit a Lasso model by calling glmnet(). The first
argument to glmnet() is the design matrix and the second
argument is a regular vector for the response.
Train a Lasso model for the model 3 and model 6 formulations,
assign the results to lasso_03 and lasso_06,
respectively.
lasso_03 <- glmnet(X03_glmnet, df$y)
lasso_06 <- glmnet(X06_glmnet, df$y)
Plot the coefficient path for each Lasso model by calling the
plot() function on the glmnet model object.
Specify the xvar argument to be 'lambda' in
the plot() call.
plot(lasso_03, xvar = 'lambda')
plot(lasso_06, xvar = 'lambda')
Now that you have visualized the coefficient path, it’s time to
identify the best 'lambda' value to use! The
cv.glmnet() function will by default use 10-fold
cross-validation to tune 'lambda'. The first argument to
cv.glmnet() is the design matrix and the second argument is
the regular vector for the response.
Tune the Lasso regularization strength with cross-validation
using the cv.glmnet() function for each model formulation.
Assign the model 3 result to lasso_03_cv_tune and assign
the model 6 result to lasso_06_cv_tune. Also specify the
alpha argument to be 1 to make sure the Lasso penalty is
applied in the cv.glmnet() call.
HINT: The random seed was assigned for you in two separate code chunks below. This will help ensure you can reproduce the cross-validation results.
### the random seed is set for you
set.seed(812312)
### tune the model 3 formulation
lasso_03_cv_tune <- cv.glmnet(X03_glmnet, df$y, alpha = 1, nfolds = 10)
### the random seed is set for you
set.seed(812312)
### tune the model 6 formulation
lasso_06_cv_tune <- cv.glmnet(X06_glmnet, df$y, alpha = 1, nfolds = 10)
Plot the cross-validation results using the default plot method for each cross-validation result. How many coefficients are remaining after tuning?
plot(lasso_03_cv_tune)
lasso_03_coef <- coef(lasso_03_cv_tune, s = "lambda.min")
num_coef_03 <- sum(lasso_03_coef != 0)-1
num_coef_03
## [1] 7
Apart from the intercept,there are 7 coefficients remaining after tuning in model 3.
plot(lasso_06_cv_tune)
lasso_06_coef <- coef(lasso_06_cv_tune, s = "lambda.min")
num_coef_06 <- sum(lasso_06_coef != 0)-1
num_coef_06
## [1] 2
Apart from the intercept,there are 2 coefficients remaining after tuning in model 6.
Which features have NOT been “turned off” by the Lasso
penalty? Use the coef() function to display the lasso model
cross-validation results to show the tuned penalized regression
coefficients for each model.
Are the final tuned models different from each
other?
coef(lasso_03_cv_tune)
## 9 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) 0.3009999
## x1 .
## I(x1^2) .
## x2 .
## I(x2^2) -0.3040403
## x1:x2 .
## x1:I(x2^2) .
## I(x1^2):x2 .
## I(x1^2):I(x2^2) .
A dot indicates that the coefficient has been turned off. Apart from the intercept only the square of x2 remains.
coef(lasso_06_cv_tune)
## 65 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) 0.2818858
## splines::ns(x1, 12)1 .
## splines::ns(x1, 12)2 .
## splines::ns(x1, 12)3 .
## splines::ns(x1, 12)4 .
## splines::ns(x1, 12)5 .
## splines::ns(x1, 12)6 .
## splines::ns(x1, 12)7 .
## splines::ns(x1, 12)8 .
## splines::ns(x1, 12)9 .
## splines::ns(x1, 12)10 .
## splines::ns(x1, 12)11 .
## splines::ns(x1, 12)12 .
## x2 .
## I(x2^2) -0.2847331
## I(x2^3) .
## I(x2^4) .
## splines::ns(x1, 12)1:x2 .
## splines::ns(x1, 12)2:x2 .
## splines::ns(x1, 12)3:x2 .
## splines::ns(x1, 12)4:x2 .
## splines::ns(x1, 12)5:x2 .
## splines::ns(x1, 12)6:x2 .
## splines::ns(x1, 12)7:x2 .
## splines::ns(x1, 12)8:x2 .
## splines::ns(x1, 12)9:x2 .
## splines::ns(x1, 12)10:x2 .
## splines::ns(x1, 12)11:x2 .
## splines::ns(x1, 12)12:x2 .
## splines::ns(x1, 12)1:I(x2^2) .
## splines::ns(x1, 12)2:I(x2^2) .
## splines::ns(x1, 12)3:I(x2^2) .
## splines::ns(x1, 12)4:I(x2^2) .
## splines::ns(x1, 12)5:I(x2^2) .
## splines::ns(x1, 12)6:I(x2^2) .
## splines::ns(x1, 12)7:I(x2^2) .
## splines::ns(x1, 12)8:I(x2^2) .
## splines::ns(x1, 12)9:I(x2^2) .
## splines::ns(x1, 12)10:I(x2^2) .
## splines::ns(x1, 12)11:I(x2^2) .
## splines::ns(x1, 12)12:I(x2^2) .
## splines::ns(x1, 12)1:I(x2^3) .
## splines::ns(x1, 12)2:I(x2^3) .
## splines::ns(x1, 12)3:I(x2^3) .
## splines::ns(x1, 12)4:I(x2^3) .
## splines::ns(x1, 12)5:I(x2^3) .
## splines::ns(x1, 12)6:I(x2^3) .
## splines::ns(x1, 12)7:I(x2^3) .
## splines::ns(x1, 12)8:I(x2^3) .
## splines::ns(x1, 12)9:I(x2^3) .
## splines::ns(x1, 12)10:I(x2^3) .
## splines::ns(x1, 12)11:I(x2^3) .
## splines::ns(x1, 12)12:I(x2^3) .
## splines::ns(x1, 12)1:I(x2^4) .
## splines::ns(x1, 12)2:I(x2^4) .
## splines::ns(x1, 12)3:I(x2^4) .
## splines::ns(x1, 12)4:I(x2^4) .
## splines::ns(x1, 12)5:I(x2^4) .
## splines::ns(x1, 12)6:I(x2^4) .
## splines::ns(x1, 12)7:I(x2^4) .
## splines::ns(x1, 12)8:I(x2^4) .
## splines::ns(x1, 12)9:I(x2^4) .
## splines::ns(x1, 12)10:I(x2^4) .
## splines::ns(x1, 12)11:I(x2^4) .
## splines::ns(x1, 12)12:I(x2^4) .
A dot indicates that the coefficient has been turned off. Apart from the intercept,only the square of x2 remains. So the final tuned models are the same.